lst:=[
# ::Package:: 

# ::Title:: 
#Integration problems of the form 
#Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+b Sec[c+d x])^n when a^2=b^2


# ::Section:: 
#Sec[c+d x]^m (a+a Sec[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (a+a Sec[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^4*(a + a*sec(c + d*x))^2, x, 6, (3*a^2*arctanh(sin(c + d*x)))/(4*d) + (6*a^2*tan(c + d*x))/(5*d) + (3*a^2*sec(c + d*x)*tan(c + d*x))/(4*d) + (3*a^2*sec(c + d*x)^2*tan(c + d*x))/(5*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(2*d) + (a^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^3*(a + a*sec(c + d*x))^2, x, 5, (7*a^2*arctanh(sin(c + d*x)))/(8*d) + (4*a^2*tan(c + d*x))/(3*d) + (7*a^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^2, x, 4, (a^2*arctanh(sin(c + d*x)))/d + (5*a^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)*tan(c + d*x))/d + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^2, x, 3, (3*a^2*arctanh(sin(c + d*x)))/(2*d) + (2*a^2*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^2, x, 2, a^2*x + (2*a^2*arctanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^2, x, 3, 2*a^2*x + (a^2*arctanh(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^2, x, 3, (3*a^2*x)/2 + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^2, x, 4, a^2*x + (5*a^2*sin(c + d*x))/(3*d) + (a^2*cos(c + d*x)*sin(c + d*x))/d + (a^2*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^2, x, 5, (7*a^2*x)/8 + (4*a^2*sin(c + d*x))/(3*d) + (7*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (2*a^2*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^2, x, 6, (3*a^2*x)/4 + (6*a^2*sin(c + d*x))/(5*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (3*a^2*cos(c + d*x)^2*sin(c + d*x))/(5*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d) + (a^2*cos(c + d*x)^4*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^3, x, 6, (13*a^3*arctanh(sin(c + d*x)))/(8*d) + (38*a^3*tan(c + d*x))/(15*d) + (13*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (19*a^3*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (11*a^3*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a^2*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^3, x, 5, (15*a^3*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*tan(c + d*x))/(2*d) + (5*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d) + ((a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^3, x, 4, (5*a^3*arctanh(sin(c + d*x)))/(2*d) + (10*a^3*tan(c + d*x))/(3*d) + (5*a^3*sec(c + d*x)*tan(c + d*x))/(6*d) + (a*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^3, x, 3, a^3*x + (7*a^3*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*tan(c + d*x))/(2*d) + (a^2*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^3, x, 4, 3*a^3*x + (3*a^3*arctanh(sin(c + d*x)))/d + (a^2*(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^3, x, 4, (7*a^3*x)/2 + (a^3*arctanh(sin(c + d*x)))/d + (5*a^3*sin(c + d*x))/(2*d) + (a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^3, x, 4, (5*a^3*x)/2 + (10*a^3*sin(c + d*x))/(3*d) + (5*a^3*cos(c + d*x)*sin(c + d*x))/(6*d) + (a*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^3, x, 5, (15*a^3*x)/8 + (5*a^3*sin(c + d*x))/(2*d) + (5*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(a + a*cos(c + d*x))^2*sin(c + d*x))/(4*d) + ((a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^3, x, 6, (13*a^3*x)/8 + (38*a^3*sin(c + d*x))/(15*d) + (13*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (19*a^3*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (11*a^3*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (a^2*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^3, x, 7, (23*a^3*x)/16 + (34*a^3*sin(c + d*x))/(15*d) + (23*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (17*a^3*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (23*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (13*a^3*cos(c + d*x)^4*sin(c + d*x))/(30*d) + (a^2*cos(c + d*x)^5*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^4, x, 7, (49*a^4*arctanh(sin(c + d*x)))/(16*d) + (24*a^4*tan(c + d*x))/(5*d) + (49*a^4*sec(c + d*x)*tan(c + d*x))/(16*d) + (12*a^4*sec(c + d*x)^2*tan(c + d*x))/(5*d) + (43*a^4*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (7*a^3*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (a^2*sec(c + d*x)^3*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^4, x, 6, (7*a^4*arctanh(sin(c + d*x)))/(2*d) + (14*a^4*tan(c + d*x))/(3*d) + (7*a^4*sec(c + d*x)*tan(c + d*x))/(6*d) + (7*a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d) + ((a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^4, x, 5, (35*a^4*arctanh(sin(c + d*x)))/(8*d) + (35*a^4*tan(c + d*x))/(6*d) + (35*a^4*sec(c + d*x)*tan(c + d*x))/(24*d) + (7*a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (a*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^4, x, 4, a^4*x + (6*a^4*arctanh(sin(c + d*x)))/d + (5*a^4*tan(c + d*x))/d + (4*a^3*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^4, x, 5, 4*a^4*x + (13*a^4*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*sin(c + d*x))/(2*d) + (3*a^3*(a + a*sec(c + d*x))*sin(c + d*x))/d + (a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^4, x, 5, (13*a^4*x)/2 + (4*a^4*arctanh(sin(c + d*x)))/d + (5*a^4*sin(c + d*x))/(2*d) + (a^3*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (a^2*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^4, x, 5, 6*a^4*x + (a^4*arctanh(sin(c + d*x)))/d + (5*a^4*sin(c + d*x))/d + (4*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^4, x, 5, (35*a^4*x)/8 + (35*a^4*sin(c + d*x))/(6*d) + (35*a^4*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (a*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^4, x, 6, (7*a^4*x)/2 + (14*a^4*sin(c + d*x))/(3*d) + (7*a^4*cos(c + d*x)*sin(c + d*x))/(6*d) + (7*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d) + ((a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^4, x, 7, (49*a^4*x)/16 + (24*a^4*sin(c + d*x))/(5*d) + (49*a^4*cos(c + d*x)*sin(c + d*x))/(16*d) + (12*a^4*cos(c + d*x)^2*sin(c + d*x))/(5*d) + (43*a^4*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (7*a^3*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (a^2*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(a + a*sec(c + d*x))^4, x, 8, (11*a^4*x)/4 + (454*a^4*sin(c + d*x))/(105*d) + (11*a^4*cos(c + d*x)*sin(c + d*x))/(4*d) + (227*a^4*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (11*a^4*cos(c + d*x)^3*sin(c + d*x))/(6*d) + (89*a^4*cos(c + d*x)^4*sin(c + d*x))/(105*d) + (8*a^3*cos(c + d*x)^5*(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (a^2*cos(c + d*x)^6*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^5/(a + a*sec(c + d*x)), x, 5, -((3*arctanh(sin(c + d*x)))/(2*a*d)) + (8*tan(c + d*x))/(3*a*d) - (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) + (4*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) - (sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x)), x, 4, (3*arctanh(sin(c + d*x)))/(2*a*d) - (2*tan(c + d*x))/(a*d) + (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) - (sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x)), x, 3, -(arctanh(sin(c + d*x))/(a*d)) + (2*tan(c + d*x))/(a*d) - (sec(c + d*x)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x)), x, 3, arctanh(sin(c + d*x))/(a*d) - tan(c + d*x)/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/(a + a*sec(c + d*x)), x, 1, tan(c + d*x)/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0/(a + a*sec(c + d*x)), x, 2, x/a - tan(c + d*x)/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1/(a + a*sec(c + d*x)), x, 3, -(x/a) + (2*sin(c + d*x))/(a*d) - sin(c + d*x)/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x)), x, 4, (3*x)/(2*a) - (2*sin(c + d*x))/(a*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3/(a + a*sec(c + d*x)), x, 5, -((3*x)/(2*a)) + (8*sin(c + d*x))/(3*a*d) - (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) + (4*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) - (cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^4/(a + a*sec(c + d*x)), x, 6, (15*x)/(8*a) - (8*sin(c + d*x))/(3*a*d) + (15*cos(c + d*x)*sin(c + d*x))/(8*a*d) - (4*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) + (5*cos(c + d*x)^3*sin(c + d*x))/(4*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^5/(a + a*sec(c + d*x))^2, x, 5, (7*arctanh(sin(c + d*x)))/(2*a^2*d) - (16*tan(c + d*x))/(3*a^2*d) + (7*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - (sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (8*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^2, x, 4, -((2*arctanh(sin(c + d*x)))/(a^2*d)) + (10*tan(c + d*x))/(3*a^2*d) - (sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*sec(c + d*x)*tan(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 3, arctanh(sin(c + d*x))/(a^2*d) - (sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (4*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 2, -(tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2)) + (2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 2, tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2) + tan(c + d*x)/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^2, x, 3, x/a^2 - tan(c + d*x)/(3*d*(a + a*sec(c + d*x))^2) - (4*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^2, x, 4, -((2*x)/a^2) + (10*sin(c + d*x))/(3*a^2*d) - sin(c + d*x)/(3*d*(a + a*sec(c + d*x))^2) - (2*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x))^2, x, 5, (7*x)/(2*a^2) - (16*sin(c + d*x))/(3*a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (8*cos(c + d*x)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3/(a + a*sec(c + d*x))^2, x, 6, -((5*x)/a^2) + (8*sin(c + d*x))/(a^2*d) - (5*cos(c + d*x)*sin(c + d*x))/(a^2*d) + (4*cos(c + d*x)^2*sin(c + d*x))/(a^2*d) - (cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (10*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^6/(a + a*sec(c + d*x))^3, x, 6, (13*arctanh(sin(c + d*x)))/(2*a^3*d) - (152*tan(c + d*x))/(15*a^3*d) + (13*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - (sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (11*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (76*sec(c + d*x)^2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^5/(a + a*sec(c + d*x))^3, x, 5, -((3*arctanh(sin(c + d*x)))/(a^3*d)) + (24*tan(c + d*x))/(5*a^3*d) - (sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (3*sec(c + d*x)^2*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (3*sec(c + d*x)*tan(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^3, x, 4, arctanh(sin(c + d*x))/(a^3*d) - (sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (7*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (22*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^3, x, 3, (sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (4*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 3, -(tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3)) + tan(c + d*x)/(5*a*d*(a + a*sec(c + d*x))^2) + tan(c + d*x)/(5*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^3, x, 3, tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^3, x, 4, x/a^3 - tan(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) - (7*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (22*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^3, x, 5, -((3*x)/a^3) + (24*sin(c + d*x))/(5*a^3*d) - sin(c + d*x)/(5*d*(a + a*sec(c + d*x))^3) - (3*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (3*sin(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x))^3, x, 6, (13*x)/(2*a^3) - (152*sin(c + d*x))/(15*a^3*d) + (13*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (11*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (76*cos(c + d*x)*sin(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^7/(a + a*sec(c + d*x))^4, x, 7, (21*arctanh(sin(c + d*x)))/(2*a^4*d) - (576*tan(c + d*x))/(35*a^4*d) + (21*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - (sec(c + d*x)^5*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*sec(c + d*x)^4*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - (43*sec(c + d*x)^3*tan(c + d*x))/(35*a^2*d*(a + a*sec(c + d*x))^2) - (288*sec(c + d*x)^2*tan(c + d*x))/(35*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^6/(a + a*sec(c + d*x))^4, x, 6, -((4*arctanh(sin(c + d*x)))/(a^4*d)) + (664*tan(c + d*x))/(105*a^4*d) - (sec(c + d*x)^4*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (12*sec(c + d*x)^3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (88*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*sec(c + d*x)*tan(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^5/(a + a*sec(c + d*x))^4, x, 5, arctanh(sin(c + d*x))/(a^4*d) - (sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*sec(c + d*x)^2*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^3) - (11*sec(c + d*x)*tan(c + d*x))/(21*a^2*d*(a + a*sec(c + d*x))^2) - (32*tan(c + d*x))/(21*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^4, x, 4, (sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + (3*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (2*tan(c + d*x))/(35*a^2*d*(a + a*sec(c + d*x))^2) + (4*tan(c + d*x))/(35*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^4, x, 4, -((sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) + (4*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (8*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (16*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^4, x, 4, -(tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4)) + (4*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (8*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (8*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^4, x, 4, tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) + (3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(35*a^2*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(35*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^4, x, 5, x/a^4 - tan(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) - (2*tan(c + d*x))/(7*a*d*(a + a*sec(c + d*x))^3) - (11*tan(c + d*x))/(21*a^2*d*(a + a*sec(c + d*x))^2) - (32*tan(c + d*x))/(21*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^4, x, 6, -((4*x)/a^4) + (664*sin(c + d*x))/(105*a^4*d) - sin(c + d*x)/(7*d*(a + a*sec(c + d*x))^4) - (12*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (88*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*sin(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x))^4, x, 7, (21*x)/(2*a^4) - (576*sin(c + d*x))/(35*a^4*d) + (21*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - (cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*cos(c + d*x)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - (43*cos(c + d*x)*sin(c + d*x))/(35*a^2*d*(a + a*sec(c + d*x))^2) - (288*cos(c + d*x)*sin(c + d*x))/(35*a^3*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^7/(a + a*sec(c + d*x))^5, x, 7, -((5*arctanh(sin(c + d*x)))/(a^5*d)) + (496*tan(c + d*x))/(63*a^5*d) - (sec(c + d*x)^5*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (5*sec(c + d*x)^4*tan(c + d*x))/(21*a*d*(a + a*sec(c + d*x))^4) - (29*sec(c + d*x)^3*tan(c + d*x))/(63*a^2*d*(a + a*sec(c + d*x))^3) - (67*sec(c + d*x)^2*tan(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) - (5*sec(c + d*x)*tan(c + d*x))/(a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^6/(a + a*sec(c + d*x))^5, x, 6, arctanh(sin(c + d*x))/(a^5*d) - (sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (13*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (34*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) - (173*sec(c + d*x)*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (488*tan(c + d*x))/(315*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^5/(a + a*sec(c + d*x))^5, x, 5, (sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) + (4*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + (4*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) - (8*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) + (16*tan(c + d*x))/(315*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^5, x, 5, -((sec(c + d*x)^4*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5)) + (5*sec(c + d*x)^3*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + (sec(c + d*x)^2*tan(c + d*x))/(21*a^2*d*(a + a*sec(c + d*x))^3) - (2*tan(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) + (4*tan(c + d*x))/(63*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^5, x, 5, -((sec(c + d*x)*tan(c + d*x))/(9*d*(a + a*sec(c + d*x))^5)) - tan(c + d*x)/(9*a*d*(a + a*sec(c + d*x))^4) + tan(c + d*x)/(15*a^2*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(45*a^3*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(45*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^5, x, 5, -(tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5)) + (5*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + tan(c + d*x)/(21*a^2*d*(a + a*sec(c + d*x))^3) + (2*tan(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) + (2*tan(c + d*x))/(63*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^5, x, 5, tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) + (4*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) + (4*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) + (8*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) + (8*tan(c + d*x))/(315*a^4*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^5, x, 6, x/a^5 - tan(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) - (13*tan(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (34*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^3) - (173*tan(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (488*tan(c + d*x))/(315*a^4*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^5, x, 7, -((5*x)/a^5) + (496*sin(c + d*x))/(63*a^5*d) - sin(c + d*x)/(9*d*(a + a*sec(c + d*x))^5) - (5*sin(c + d*x))/(21*a*d*(a + a*sec(c + d*x))^4) - (29*sin(c + d*x))/(63*a^2*d*(a + a*sec(c + d*x))^3) - (67*sin(c + d*x))/(63*a^3*d*(a + a*sec(c + d*x))^2) - (5*sin(c + d*x))/(a^4*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x))^5, x, 8, (31*x)/(2*a^5) - (7664*sin(c + d*x))/(315*a^5*d) + (31*cos(c + d*x)*sin(c + d*x))/(2*a^5*d) - (cos(c + d*x)*sin(c + d*x))/(9*d*(a + a*sec(c + d*x))^5) - (17*cos(c + d*x)*sin(c + d*x))/(63*a*d*(a + a*sec(c + d*x))^4) - (28*cos(c + d*x)*sin(c + d*x))/(45*a^2*d*(a + a*sec(c + d*x))^3) - (577*cos(c + d*x)*sin(c + d*x))/(315*a^3*d*(a + a*sec(c + d*x))^2) - (3832*cos(c + d*x)*sin(c + d*x))/(315*a^4*d*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^4*sqrt(a + a*sec(c + d*x)), x, 4, (16*a*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (12*a*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))) + (16*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(35*d)],
[sec(c + d*x)^3*sqrt(a + a*sec(c + d*x)), x, 3, (8*a*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (8*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d)],
[sec(c + d*x)^2*sqrt(a + a*sec(c + d*x)), x, 2, (2*a*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*sqrt(a + a*sec(c + d*x)), x, 1, (2*a*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*sqrt(a + a*sec(c + d*x)), x, 1, (2*sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d],
[cos(c + d*x)^1*sqrt(a + a*sec(c + d*x)), x, 2, (sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*sqrt(a + a*sec(c + d*x)), x, 3, (3*sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (3*a*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*sqrt(a + a*sec(c + d*x)), x, 4, (5*sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (5*a*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (5*a*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*sqrt(a + a*sec(c + d*x)), x, 5, (35*sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (35*a*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (35*a*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (7*a*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x)))],

[sqrt(a - a*sec(c + d*x)), x, 1, (2*sqrt(a)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/d],


[sec(c + d*x)^4*(a + a*sec(c + d*x))^(3/2), x, 6, (272*a^2*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (68*a^2*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (34*a^2*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sec(c + d*x)^4*tan(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x))) + (272*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d)],
[sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2), x, 5, (104*a^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (26*a^2*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))) + (104*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2), x, 3, (8*a^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(3/2), x, 2, (8*a^2*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(3/2), x, 3, (2*a^(3/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(3/2), x, 4, (3*a^(3/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2), x, 4, (7*a^(3/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (7*a^2*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2), x, 5, (11*a^(3/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (11*a^2*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (11*a^2*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(a + a*sec(c + d*x))^(5/2), x, 6, (1136*a^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (284*a^3*sec(c + d*x)^2*tan(c + d*x))/(231*d*sqrt(a + a*sec(c + d*x))) + (710*a^3*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (46*a^3*sec(c + d*x)^4*tan(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) + (1136*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(693*d) + (2*a^2*sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(11*d)],
[sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2), x, 5, (584*a^3*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (146*a^3*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (38*a^3*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (584*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a^2*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2), x, 4, (64*a^3*tan(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(5/2), x, 3, (64*a^3*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*a*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(5/2), x, 4, (2*a^(5/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (14*a^3*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(5/2), x, 3, (5*a^(5/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d - (a^3*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2), x, 3, (19*a^(5/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (9*a^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2), x, 4, (25*a^(5/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (25*a^3*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (13*a^3*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2), x, 5, (163*a^(5/2)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (163*a^3*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (163*a^3*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (17*a^3*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4/sqrt(a + a*sec(c + d*x)), x, 4, -((sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (26*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) - (2*sec(c + d*x)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*sec(c + d*x)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/sqrt(a + a*sec(c + d*x)), x, 2, -((sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1/sqrt(a + a*sec(c + d*x)), x, 1, (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)],
[sec(c + d*x)^0/sqrt(a + a*sec(c + d*x)), x, 3, (2*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^1/sqrt(a + a*sec(c + d*x)), x, 4, -(arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + sin(c + d*x)/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/sqrt(a + a*sec(c + d*x)), x, 5, (7*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - sin(c + d*x)/(4*d*sqrt(a + a*sec(c + d*x))) + (cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],

[sec(x)/sqrt(a + a*sec(x)), x, 1, (sqrt(2)*arctan((sqrt(a)*tan(x))/(sqrt(2)*sqrt(a + a*sec(x)))))/sqrt(a)],


[sec(c + d*x)/sqrt(a - a*sec(c + d*x)), x, 1, -((sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d))],
[1/sqrt(a - a*sec(c + d*x)), x, 3, (2*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d)],


[sec(c + d*x)^5/(a + a*sec(c + d*x))^(3/2), x, 5, -((15*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - (sec(c + d*x)^3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (49*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) - (13*sec(c + d*x)*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) + (9*sec(c + d*x)^2*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^(3/2), x, 4, (11*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (19*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x))) + (7*sec(c + d*x)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^(3/2), x, 3, -((7*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - (sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (5*tan(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^(3/2), x, 2, (3*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^(3/2), x, 2, arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^(3/2), x, 4, (2*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - (5*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^(3/2), x, 5, -((3*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d)) + (9*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*sec(c + d*x))^(3/2)) + (3*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2/(a + a*sec(c + d*x))^(3/2), x, 6, (19*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(3/2)*d) - (13*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (7*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + (cos(c + d*x)*sin(c + d*x))/(a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^5/(a + a*sec(c + d*x))^(5/2), x, 5, (163*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)^3*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (17*sec(c + d*x)^2*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (299*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x))) + (95*sec(c + d*x)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^4/(a + a*sec(c + d*x))^(5/2), x, 4, -((75*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - (sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (13*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (49*tan(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3/(a + a*sec(c + d*x))^(5/2), x, 3, (19*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (9*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^2/(a + a*sec(c + d*x))^(5/2), x, 3, (5*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) + (5*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^1/(a + a*sec(c + d*x))^(5/2), x, 3, (3*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) + (3*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0/(a + a*sec(c + d*x))^(5/2), x, 5, (2*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - (43*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - tan(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) - (11*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1/(a + a*sec(c + d*x))^(5/2), x, 6, -((5*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d)) + (115*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*sec(c + d*x))^(5/2)) - (15*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[sec(a + b*x)^(9/2), x, 4, (10*sqrt(cos(a + b*x))*EllipticF((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(21*b) + (10*sec(a + b*x)^(3/2)*sin(a + b*x))/(21*b) + (2*sec(a + b*x)^(7/2)*sin(a + b*x))/(7*b)],
[sec(a + b*x)^(7/2), x, 4, -((6*sqrt(cos(a + b*x))*EllipticE((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(5*b)) + (6*sqrt(sec(a + b*x))*sin(a + b*x))/(5*b) + (2*sec(a + b*x)^(5/2)*sin(a + b*x))/(5*b)],
[sec(a + b*x)^(5/2), x, 3, (2*sqrt(cos(a + b*x))*EllipticF((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(3*b) + (2*sec(a + b*x)^(3/2)*sin(a + b*x))/(3*b)],
[sec(a + b*x)^(3/2), x, 3, -((2*sqrt(cos(a + b*x))*EllipticE((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/b) + (2*sqrt(sec(a + b*x))*sin(a + b*x))/b],
[sec(a + b*x)^(1/2), x, 2, (2*sqrt(cos(a + b*x))*EllipticF((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/b],
[1/sec(a + b*x)^(1/2), x, 2, (2*sqrt(cos(a + b*x))*EllipticE((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/b],
[1/sec(a + b*x)^(3/2), x, 3, (2*sqrt(cos(a + b*x))*EllipticF((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(3*b) + (2*sin(a + b*x))/(3*b*sqrt(sec(a + b*x)))],
[1/sec(a + b*x)^(5/2), x, 3, (6*sqrt(cos(a + b*x))*EllipticE((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(5*b) + (2*sin(a + b*x))/(5*b*sec(a + b*x)^(3/2))],
[1/sec(a + b*x)^(7/2), x, 4, (10*sqrt(cos(a + b*x))*EllipticF((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(21*b) + (2*sin(a + b*x))/(7*b*sec(a + b*x)^(5/2)) + (10*sin(a + b*x))/(21*b*sqrt(sec(a + b*x)))],
[1/sec(a + b*x)^(9/2), x, 4, (14*sqrt(cos(a + b*x))*EllipticE((1/2)*(a + b*x), 2)*sqrt(sec(a + b*x)))/(15*b) + (2*sin(a + b*x))/(9*b*sec(a + b*x)^(7/2)) + (14*sin(a + b*x))/(45*b*sec(a + b*x)^(3/2))],


[(a + a*sec(c + d*x))*sec(c + d*x)^(5/2), x, 7, (-6*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (6*a*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*sec(c + d*x))*sec(c + d*x)^(3/2), x, 6, (-2*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*sec(c + d*x))*sec(c + d*x)^(1/2), x, 5, (-2*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*sec(c + d*x))/sec(c + d*x)^(1/2), x, 4, (2*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d],
[(a + a*sec(c + d*x))/sec(c + d*x)^(3/2), x, 5, (2*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))/sec(c + d*x)^(5/2), x, 6, (6*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))/sec(c + d*x)^(7/2), x, 7, (6*a*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (10*a*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*a*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + a*sec(c + d*x))^2*sec(c + d*x)^(5/2), x, 8, (-12*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(7*d) + (12*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(7*d) + (4*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*sec(c + d*x))^2*sec(c + d*x)^(3/2), x, 7, (-16*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*sec(c + d*x))^2*sec(c + d*x)^(1/2), x, 6, (-4*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*sec(c + d*x))^2/sec(c + d*x)^(1/2), x, 4, (4*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*sec(c + d*x))^2/sec(c + d*x)^(3/2), x, 5, (4*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^2/sec(c + d*x)^(5/2), x, 6, (16*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^2/sec(c + d*x)^(7/2), x, 7, (12*a^2*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(7*d) + (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (8*a^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],


[(a + a*sec(c + d*x))^3*sec(c + d*x)^(3/2), x, 8, -((28*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (52*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (28*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (52*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (32*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a^2*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[(a + a*sec(c + d*x))^3*sec(c + d*x)^(1/2), x, 7, -((36*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (36*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[(a + a*sec(c + d*x))^3/sec(c + d*x)^(1/2), x, 6, -((4*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (20*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[(a + a*sec(c + d*x))^3/sec(c + d*x)^(3/2), x, 6, (4*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (20*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^3/sec(c + d*x)^(5/2), x, 6, (36*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (8*a^3*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^3/sec(c + d*x)^(7/2), x, 7, (28*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (52*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (32*a^3*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (52*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^3/sec(c + d*x)^(9/2), x, 8, (68*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (44*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (40*a^3*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (68*a^3*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (44*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[(a + a*sec(c + d*x))^4*sec(c + d*x)^(3/2), x, 9, -((152*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (152*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (32*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(7*d) + (188*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (44*a^3*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*a^2*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[(a + a*sec(c + d*x))^4*sec(c + d*x)^(1/2), x, 8, -((64*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (136*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (64*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (332*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (36*a^3*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a^2*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(1/2), x, 7, -((56*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (164*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (28*a^3*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(3/2), x, 6, (40*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (20*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(5/2), x, 7, (56*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (28*a^3*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(7/2), x, 7, (64*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (136*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (332*a^4*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (36*a^3*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(9/2), x, 8, (152*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (188*a^4*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (32*a^4*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (44*a^3*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[(a + a*sec(c + d*x))^4/sec(c + d*x)^(11/2), x, 9, (128*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (904*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (860*a^4*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (128*a^4*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (904*a^4*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (52*a^3*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*a^2*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(7/2)/(a + a*sec(c + d*x)), x, 7, (3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) - (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)/(a + a*sec(c + d*x)), x, 6, -((3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*sec(c + d*x)), x, 5, (sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*sec(c + d*x)), x, 5, -((sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))), x, 5, (3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, -((3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (21*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d) - (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (7*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^2, x, 8, (7*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (10*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (7*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (10*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (7*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^2, x, 7, -((4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) - (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^2, x, 6, (sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (sqrt(sec(c + d*x))*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^2, x, 3, (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)],
[sec(c + d*x)^(1/2)/(a + a*sec(c + d*x))^2, x, 6, -((sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2), x, 6, (4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) - (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (5*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 7, -((7*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (10*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - (7*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 8, (56*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (56*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(a^2*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (3*sin(c + d*x))/(a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[sec(c + d*x)^(11/2)/(a + a*sec(c + d*x))^3, x, 9, (119*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (11*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (119*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + (11*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a^3*d) - (sec(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - (119*sec(c + d*x)^(5/2)*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^3, x, 8, -((49*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) - (13*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^3, x, 7, (9*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (9*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^3, x, 7, (sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^3, x, 7, -((sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*sec(c + d*x))^3, x, 7, -((9*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(2*a^2*d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3), x, 7, (49*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (13*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 8, -((119*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (11*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - (119*sin(c + d*x))/(30*a^2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 9, (231*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (21*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (77*sin(c + d*x))/(10*a^3*d*sec(c + d*x)^(3/2)) - (21*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(5*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - (4*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (63*sin(c + d*x))/(10*a^2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (a+a Sec[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x)), x, 4, (3*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (3*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x)), x, 2, (2*sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/sqrt(sec(c + d*x)), x, 1, (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(3/2), x, 2, (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(5/2), x, 3, (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/sec(c + d*x)^(7/2), x, 4, (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (12*a*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (16*a*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (32*a*sqrt(sec(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2), x, 6, (11*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (11*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (11*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2), x, 5, (7*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (7*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 4, (3*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/sqrt(sec(c + d*x)), x, 3, (2*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(3/2), x, 2, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(5/2), x, 3, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(7/2), x, 5, (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (26*a^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (104*a^2*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (208*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/sec(c + d*x)^(9/2), x, 6, (2*a^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (34*a^2*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (68*a^2*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (272*a^2*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (544*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2), x, 6, (163*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (163*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (163*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (17*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2), x, 5, (25*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (25*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (13*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (19*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (9*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[(a + a*sec(c + d*x))^(5/2)/sqrt(sec(c + d*x)), x, 4, (5*sqrt(2)*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(3/2), x, 4, (2*sqrt(2)*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (14*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(5/2), x, 3, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(7/2), x, 4, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(9/2), x, 5, (38*a^3*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (146*a^3*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (584*a^3*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (1168*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[(a + a*sec(c + d*x))^(5/2)/sec(c + d*x)^(11/2), x, 6, (46*a^3*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (710*a^3*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (284*a^3*sin(c + d*x))/(231*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (1136*a^3*sin(c + d*x))/(693*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2272*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


[sec(c + d*x)^(-1/4)*(a + a*sec(c + d*x))^(3/2), x, 2, (4*a^2*sec(c + d*x)^(3/4)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(5/2)/sqrt(1 + sec(c + d*x)), x, 6, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) - (sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(1 + sec(c + d*x)))],
[sec(c + d*x)^(3/2)/sqrt(1 + sec(c + d*x)), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x)))],
[sqrt(sec(c + d*x))/sqrt(1 + sec(c + d*x)), x, 2, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x)))],
[1/(sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))), x, 3, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(1 + sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*sqrt(1 + sec(c + d*x))), x, 4, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) + (2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(1 + sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*sqrt(1 + sec(c + d*x))), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(1 + sec(c + d*x))) + (2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(1 + sec(c + d*x))) - (2*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(1 + sec(c + d*x))) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(1 + sec(c + d*x)))],


[sec(c + d*x)^(5/2)/sqrt(a + a*sec(c + d*x)), x, 6, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) - (sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/sqrt(a + a*sec(c + d*x)), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))],
[sqrt(sec(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 2, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))],
[1/(sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 3, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 4, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (2*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^(3/2), x, 7, (9*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - (3*sqrt(2)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (3*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(3/2), x, 6, -((5*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(3/2), x, 3, (arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sqrt(sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 3, (3*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[1/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 4, -((7*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) - (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (5*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 5, (11*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (7*sin(c + d*x))/(6*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, -((15*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) - sin(c + d*x)/(2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + (9*sin(c + d*x))/(10*a*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (13*sin(c + d*x))/(10*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(9/2)/(a + a*sec(c + d*x))^(5/2), x, 8, (115*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - (5*sqrt(2)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (15*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (35*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(7/2)/(a + a*sec(c + d*x))^(5/2), x, 7, -((43*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (11*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(5/2), x, 4, (3*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (3*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(5/2), x, 4, (5*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sqrt(sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 5, (19*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (13*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)), (19*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + (5*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (9*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[1/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, -((75*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))) - (sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[1/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 6, (163*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - (17*sin(c + d*x))/(16*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (95*sin(c + d*x))/(48*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (299*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/3) (a+a Sec[c+d x])^(n/3)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(1/3), x, 2, Int(sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(1/3), x)],
[sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(2/3), x, 1, Int(sec(c + d*x)^(4/3)*(a + a*sec(c + d*x))^(2/3), x)],
[sec(c + d*x)^(5/3)*(a + a*sec(c + d*x))^(2/3), x, 2, Int(sec(c + d*x)^(5/3)*(a + a*sec(c + d*x))^(2/3), x)],


# ::Subsubsection:: 
#n<0


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)), x, 6, (6*a*EllipticE((c + d*x)/2, 2))/(5*d) + (10*a*EllipticF((c + d*x)/2, 2))/(21*d) + (10*a*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)), x, 5, (6*a*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)), x, 4, (2*a*EllipticE((c + d*x)/2, 2))/d + (2*a*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + a*sec(c + d*x)), x, 3, (2*a*EllipticE((c + d*x)/2, 2))/d + (2*a*EllipticF((c + d*x)/2, 2))/d],
[(a + a*sec(c + d*x))/cos(c + d*x)^(1/2), x, 4, (-2*a*EllipticE((c + d*x)/2, 2))/d + (2*a*EllipticF((c + d*x)/2, 2))/d + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))/cos(c + d*x)^(3/2), x, 5, (-2*a*EllipticE((c + d*x)/2, 2))/d + (2*a*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))/cos(c + d*x)^(5/2), x, 6, (-6*a*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*a*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))/cos(c + d*x)^(7/2), x, 7, (-6*a*EllipticE((c + d*x)/2, 2))/(5*d) + (10*a*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (10*a*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (6*a*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2, x, 7, (32*a^2*EllipticE((c + d*x)/2, 2))/(15*d) + (20*a^2*EllipticF((c + d*x)/2, 2))/(21*d) + (20*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2, x, 6, (12*a^2*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*EllipticF((c + d*x)/2, 2))/(7*d) + (8*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (4*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2, x, 5, (16*a^2*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2, x, 4, (4*a^2*EllipticE((c + d*x)/2, 2))/d + (8*a^2*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2, x, 3, (4*a^2*EllipticF((c + d*x)/2, 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))^2/cos(c + d*x)^(1/2), x, 5, (-4*a^2*EllipticE((c + d*x)/2, 2))/d + (8*a^2*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))^2/cos(c + d*x)^(3/2), x, 6, (-16*a^2*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (16*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))^2/cos(c + d*x)^(5/2), x, 7, (-12*a^2*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*EllipticF((c + d*x)/2, 2))/(7*d) + (2*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (4*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (8*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2)) + (12*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3, x, 7, (68*a^3*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (44*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (44*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (68*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (40*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3, x, 6, (28*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (52*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (52*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*a^2*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3, x, 5, (36*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*EllipticF((1/2)*(c + d*x), 2))/d + (8*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3, x, 5, (4*a^3*EllipticE((1/2)*(c + d*x), 2))/d + (20*a^3*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (4*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3, x, 5, -((4*a^3*EllipticE((1/2)*(c + d*x), 2))/d) + (20*a^3*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (16*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))^3/cos(c + d*x)^(1/2), x, 6, -((36*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*EllipticF((1/2)*(c + d*x), 2))/d + (8*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (36*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^3/cos(c + d*x)^(3/2), x, 7, -((28*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (52*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (32*a^3*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (52*a^3*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (28*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)/(a + a*sec(c + d*x)), x, 6, (21*EllipticE((1/2)*(c + d*x), 2))/(5*a*d) - (5*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*sec(c + d*x)), x, 5, -((3*EllipticE((1/2)*(c + d*x), 2))/(a*d)) + (5*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*sec(c + d*x)), x, 4, (3*EllipticE((1/2)*(c + d*x), 2))/(a*d) - EllipticF((1/2)*(c + d*x), 2)/(a*d) - sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))), x, 4, -(EllipticE((1/2)*(c + d*x), 2)/(a*d)) + EllipticF((1/2)*(c + d*x), 2)/(a*d) + sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 4, EllipticE((1/2)*(c + d*x), 2)/(a*d) + EllipticF((1/2)*(c + d*x), 2)/(a*d) - sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 5, -((3*EllipticE((1/2)*(c + d*x), 2))/(a*d)) - EllipticF((1/2)*(c + d*x), 2)/(a*d) + (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))), x, 6, (3*EllipticE((1/2)*(c + d*x), 2))/(a*d) + (5*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^2, x, 7, (56*EllipticE((1/2)*(c + d*x), 2))/(5*a^2*d) - (5*EllipticF((1/2)*(c + d*x), 2))/(a^2*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d) + (56*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (3*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^2, x, 6, -((7*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) + (10*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (7*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*sec(c + d*x))^2, x, 5, (4*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) - (5*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - sin(c + d*x)/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - (5*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^2), x, 5, -(EllipticE((1/2)*(c + d*x), 2)/(a^2*d)) + (2*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + sin(c + d*x)/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + (2*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 2, EllipticF((1/2)*(c + d*x), 2)/(3*a^2*d) + sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)],
[1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 5, EllipticE((1/2)*(c + d*x), 2)/(a^2*d) + (2*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - sin(c + d*x)/(a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2), x, 6, -((4*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) - (5*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (4*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) - (5*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2), x, 7, (7*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (10*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - (7*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - (7*sin(c + d*x))/(3*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^3, x, 8, (231*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - (21*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) - (21*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) + (77*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^3*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (4*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (63*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^3, x, 7, -((119*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + (11*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - (119*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*sec(c + d*x))^3, x, 6, (49*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - (13*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - (13*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^3), x, 6, -((9*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + EllipticF((1/2)*(c + d*x), 2)/(2*a^3*d) + sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) + sin(c + d*x)/(5*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(2*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 6, -(EllipticE((1/2)*(c + d*x), 2)/(10*a^3*d)) + EllipticF((1/2)*(c + d*x), 2)/(6*a^3*d) - sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) + (2*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 6, EllipticE((1/2)*(c + d*x), 2)/(10*a^3*d) + EllipticF((1/2)*(c + d*x), 2)/(6*a^3*d) - sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - (4*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + sin(c + d*x)/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3), x, 6, (9*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + EllipticF((1/2)*(c + d*x), 2)/(2*a^3*d) - sin(c + d*x)/(5*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(5*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (9*sin(c + d*x))/(10*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3), x, 7, -((49*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) - (13*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + (49*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) - (13*sin(c + d*x))/(6*a^2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3), x, 8, (119*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + (11*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - (119*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - (119*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x)), x, 4, (32*a*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*sqrt(cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (12*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x)), x, 3, (16*a*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)), x, 2, (4*a*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)), x, 1, (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/sqrt(cos(c + d*x)), x, 2, (2*sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/cos(c + d*x)^(3/2), x, 3, (sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[sqrt(a + a*sec(c + d*x))/cos(c + d*x)^(5/2), x, 4, (3*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (3*a*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2), x, 5, (208*a^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (104*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (26*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2), x, 3, (8*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2), x, 2, (8*a^2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 3, (2*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/sqrt(cos(c + d*x)), x, 4, (3*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/cos(c + d*x)^(3/2), x, 5, (7*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (7*a^2*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)/cos(c + d*x)^(5/2), x, 6, (11*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (11*a^2*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (11*a^2*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2), x, 5, (1168*a^3*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (584*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (146*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (38*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2), x, 4, (64*a^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*a*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2), x, 3, (64*a^3*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2), x, 4, (2*sqrt(2)*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (14*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (5*sqrt(2)*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*sec(c + d*x))^(5/2)/sqrt(cos(c + d*x)), x, 4, (19*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (9*a^3*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^(5/2)/cos(c + d*x)^(3/2), x, 5, (25*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (13*a^3*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (25*a^3*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^(5/2)/cos(c + d*x)^(5/2), x, 6, (163*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (17*a^3*sin(c + d*x))/(24*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (163*a^3*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (163*a^3*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)/sqrt(a + a*sec(c + d*x)), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)/sqrt(a + a*sec(c + d*x)), x, 4, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(cos(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 3, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 2, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))), x, 7, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (7*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + a*sec(c + d*x))^(3/2), x, 6, (-15*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (49*sin(c + d*x))/(10*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (13*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) + (9*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(3/2), x, 5, (11*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - (19*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (7*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(cos(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 4, -((7*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))) - sin(c + d*x)/(2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (5*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 3, (3*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 3, (arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, (-5*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)), x, 7, (9*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (3*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + (3*sin(c + d*x))/(2*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[cos(c + d*x)^(3/2)/(a + a*sec(c + d*x))^(5/2), x, 6, (163*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - (17*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (299*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (95*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(cos(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 5, (-75*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (49*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, (19*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), (19*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) + (5*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) - (9*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[1/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 4, (5*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + (5*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 4, (3*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + (3*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, (-43*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) - (11*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[1/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)), x, 8, (115*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (5*sqrt(2)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - sin(c + d*x)/(4*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)) - (15*sin(c + d*x))/(16*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


# ::Section:: 
#Sec[c+d x]^m (A+B Sec[c+d x]) (a+a Sec[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (a+a Sec[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>=0


[sec(c + d*x)^5*(A + B*sec(c + d*x)), x, 6, (3*A*arctanh(sin(c + d*x)))/(8*d) + (8*B*tan(c + d*x))/(15*d) + (3*A*sec(c + d*x)*tan(c + d*x))/(8*d) + (4*B*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (A*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (B*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^4*(A + B*sec(c + d*x)), x, 5, (3*B*arctanh(sin(c + d*x)))/(8*d) + (2*A*tan(c + d*x))/(3*d) + (3*B*sec(c + d*x)*tan(c + d*x))/(8*d) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (B*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^3*(A + B*sec(c + d*x)), x, 4, (A*arctanh(sin(c + d*x)))/(2*d) + (2*B*tan(c + d*x))/(3*d) + (A*sec(c + d*x)*tan(c + d*x))/(2*d) + (B*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x)), x, 3, (B*arctanh(sin(c + d*x)))/(2*d) + (A*tan(c + d*x))/d + (B*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x)), x, 2, (A*arctanh(sin(c + d*x)))/d + (B*tan(c + d*x))/d],
[sec(c + d*x)^0*(A + B*sec(c + d*x)), x, 2, A*x + (B*arctanh(sin(c + d*x)))/d],
[cos(c + d*x)^1*(A + B*sec(c + d*x)), x, 2, B*x + (A*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x)), x, 3, (A*x)/2 + (B*sin(c + d*x))/d + (A*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x)), x, 4, (B*x)/2 + (2*A*sin(c + d*x))/(3*d) + (B*cos(c + d*x)*sin(c + d*x))/(2*d) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x)), x, 5, (3*A*x)/8 + (2*B*sin(c + d*x))/(3*d) + (3*A*cos(c + d*x)*sin(c + d*x))/(8*d) + (B*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x)), x, 6, (3*B*x)/8 + (8*A*sin(c + d*x))/(15*d) + (3*B*cos(c + d*x)*sin(c + d*x))/(8*d) + (4*A*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (B*cos(c + d*x)^3*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^4*sin(c + d*x))/(5*d)],


[sec(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 6, (3*a*(A + B)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(5*A + 4*B)*tan(c + d*x))/(15*d) + (3*a*(A + B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(5*A + 4*B)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a*(A + B)*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (a*B*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 5, (a*(4*A + 3*B)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(A + B)*tan(c + d*x))/(3*d) + (a*(4*A + 3*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(A + B)*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a*B*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 4, (a*(A + B)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*A + 2*B)*tan(c + d*x))/(3*d) + (a*(A + B)*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*B*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 3, (a*(2*A + B)*arctanh(sin(c + d*x)))/(2*d) + (a*(A + B)*tan(c + d*x))/d + (a*B*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 2, a*A*x + (a*(A + B)*arctanh(sin(c + d*x)))/d + (a*B*tan(c + d*x))/d],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 3, (a*A + a*B)*x + (a*B*arctanh(sin(c + d*x)))/d + (a*A*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 3, (a*(A + 2*B)*x)/2 + (a*(A + B)*sin(c + d*x))/d + (a*A*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 4, (a*(A + B)*x)/2 + (a*(2*A + 3*B)*sin(c + d*x))/(3*d) + (a*(A + B)*cos(c + d*x)*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 5, (a*(3*A + 4*B)*x)/8 + (2*a*(A + B)*sin(c + d*x))/(3*d) + (a*(3*A + 4*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(A + B)*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (a*A*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + a*sec(c + d*x)), x, 6, (3/8)*a*(A + B)*x + (2*a*(4*A + 5*B)*sin(c + d*x))/(15*d) + (3*a*(A + B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(4*A + 5*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a*(A + B)*cos(c + d*x)^3*sin(c + d*x))/(4*d) + (a*A*cos(c + d*x)^4*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 6, (a^2*(7*A + 6*B)*arctanh(sin(c + d*x)))/(8*d) + (2*a^2*(10*A + 9*B)*tan(c + d*x))/(15*d) + (a^2*(7*A + 6*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(10*A + 9*B)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a^2*(5*A + 6*B)*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*B*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 5, (a^2*(8*A + 7*B)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(5*A + 4*B)*tan(c + d*x))/(3*d) + (a^2*(8*A + 7*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(4*A + 5*B)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (a*B*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(4*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 4, (a^2*(3*A + 2*B)*arctanh(sin(c + d*x)))/(2*d) + (2*a^2*(3*A + 2*B)*tan(c + d*x))/(3*d) + (a^2*(3*A + 2*B)*sec(c + d*x)*tan(c + d*x))/(6*d) + (B*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 3, a^2*A*x + (a^2*(4*A + 3*B)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*A + 3*B)*tan(c + d*x))/(2*d) + (a*B*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 4, (2*a^2*A + a^2*B)*x + (a^2*(A + 2*B)*arctanh(sin(c + d*x)))/d + (a^2*(A - B)*sin(c + d*x))/d + (a*B*(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 4, (1/2)*a^2*(3*A + 4*B)*x + (a^2*B*arctanh(sin(c + d*x)))/d + (a^2*(3*A + 2*B)*sin(c + d*x))/(2*d) + (a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 4, (a^2*(2*A + 3*B)*x)/2 + (2*a^2*(2*A + 3*B)*sin(c + d*x))/(3*d) + (a^2*(2*A + 3*B)*cos(c + d*x)*sin(c + d*x))/(6*d) + (A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 5, (1/8)*a^2*(7*A + 8*B)*x + (a^2*(4*A + 5*B)*sin(c + d*x))/(3*d) + (a^2*(7*A + 8*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(5*A + 4*B)*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (a*A*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^2, x, 6, (1/8)*a^2*(6*A + 7*B)*x + (2*a^2*(9*A + 10*B)*sin(c + d*x))/(15*d) + (a^2*(6*A + 7*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(9*A + 10*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(6*A + 5*B)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 7, (a^3*(26*A + 23*B)*arctanh(sin(c + d*x)))/(16*d) + (2*a^3*(19*A + 17*B)*tan(c + d*x))/(15*d) + (a^3*(26*A + 23*B)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(19*A + 17*B)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a^3*(22*A + 21*B)*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (a^2*(3*A + 4*B)*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (a*B*sec(c + d*x)^3*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 6, (a^3*(15*A + 13*B)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(45*A + 38*B)*tan(c + d*x))/(15*d) + (a^3*(15*A + 13*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*(45*A + 43*B)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a^2*(5*A + 7*B)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(20*d) + (a*B*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 5, (5*a^3*(4*A + 3*B)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(4*A + 3*B)*tan(c + d*x))/(6*d) + (5*a^3*(4*A + 3*B)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(4*A + 3*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (B*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 4, a^3*A*x + (a^3*(7*A + 5*B)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(A + B)*tan(c + d*x))/(2*d) + (a^2*(3*A + 5*B)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*B*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 5, (3*a^3*A + a^3*B)*x + (a^3*(6*A + 7*B)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*B*sin(c + d*x))/(2*d) + (a^2*(A + 2*B)*(a + a*sec(c + d*x))*sin(c + d*x))/d + (a*B*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 5, (1/2)*a^3*(7*A + 6*B)*x + (a^3*(A + 3*B)*arctanh(sin(c + d*x)))/d + (5*a^3*A*sin(c + d*x))/(2*d) - (a^2*(A - 2*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 5, (a^3*(5*A + 7*B)*x)/2 + (a^3*B*arctanh(sin(c + d*x)))/d + (5*a^3*(A + B)*sin(c + d*x))/(2*d) + (a^2*(5*A + 3*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 5, (5/8)*a^3*(3*A + 4*B)*x + (5*a^3*(3*A + 4*B)*sin(c + d*x))/(6*d) + (5*a^3*(3*A + 4*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(3*A + 4*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 6, (1/8)*a^3*(13*A + 15*B)*x + (a^3*(38*A + 45*B)*sin(c + d*x))/(15*d) + (a^3*(13*A + 15*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*(43*A + 45*B)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + (a^2*(7*A + 5*B)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(20*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^3, x, 7, (1/16)*a^3*(23*A + 26*B)*x + (2*a^3*(17*A + 19*B)*sin(c + d*x))/(15*d) + (a^3*(23*A + 26*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(17*A + 19*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^3*(21*A + 22*B)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a^2*(4*A + 3*B)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (a*A*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(6*d)],


[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 7, (7*a^4*(8*A + 7*B)*arctanh(sin(c + d*x)))/(16*d) + (a^4*(83*A + 72*B)*tan(c + d*x))/(15*d) + (7*a^4*(8*A + 7*B)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^4*(176*A + 159*B)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^3*(72*A + 73*B)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(120*d) + (a^2*(2*A + 3*B)*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(10*d) + (a*B*sec(c + d*x)^2*(a + a*sec(c + d*x))^3*tan(c + d*x))/(6*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (7*a^4*(5*A + 4*B)*arctanh(sin(c + d*x)))/(8*d) + (7*a^4*(5*A + 4*B)*tan(c + d*x))/(6*d) + (7*a^4*(5*A + 4*B)*sec(c + d*x)*tan(c + d*x))/(24*d) + (7*a^2*(5*A + 4*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (a*(5*A + 4*B)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (B*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 5, a^4*A*x + (a^4*(48*A + 35*B)*arctanh(sin(c + d*x)))/(8*d) + (5*a^4*(8*A + 7*B)*tan(c + d*x))/(8*d) + (a^3*(32*A + 35*B)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a^2*(4*A + 7*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (a*B*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (4*a^4*A + a^4*B)*x + (a^4*(13*A + 12*B)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(A + 2*B)*sin(c + d*x))/(2*d) + (a^3*(9*A + 11*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (a^2*(A + 2*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*B*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (1/2)*a^4*(13*A + 8*B)*x + (a^4*(8*A + 13*B)*arctanh(sin(c + d*x)))/(2*d) + (5*a^4*(A - B)*sin(c + d*x))/(2*d) + (a^3*(A + 6*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) - (a^2*(A - B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (1/2)*a^4*(12*A + 13*B)*x + (a^4*(A + 4*B)*arctanh(sin(c + d*x)))/d + (5*a^4*(2*A + B)*sin(c + d*x))/(2*d) - (a^3*(8*A - 3*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) + (a^2*(2*A + B)*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)^2*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (1/8)*a^4*(35*A + 48*B)*x + (a^4*B*arctanh(sin(c + d*x)))/d + (5*a^4*(7*A + 8*B)*sin(c + d*x))/(8*d) + (a^3*(35*A + 32*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(7*A + 4*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (a*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 6, (7/8)*a^4*(4*A + 5*B)*x + (7*a^4*(4*A + 5*B)*sin(c + d*x))/(6*d) + (7*a^4*(4*A + 5*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a^2*(4*A + 5*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (a*(4*A + 5*B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (A*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 7, (7/16)*a^4*(7*A + 8*B)*x + (a^4*(72*A + 83*B)*sin(c + d*x))/(15*d) + (7*a^4*(7*A + 8*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^4*(159*A + 176*B)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + (a^3*(73*A + 72*B)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(120*d) + (a^2*(3*A + 2*B)*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(10*d) + (a*A*cos(c + d*x)^5*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^4, x, 8, (1/16)*a^4*(44*A + 49*B)*x + (2*a^4*(227*A + 252*B)*sin(c + d*x))/(105*d) + (a^4*(44*A + 49*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^4*(227*A + 252*B)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^4*(276*A + 301*B)*cos(c + d*x)^3*sin(c + d*x))/(280*d) + (7*a^3*(A + B)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (a^2*(10*A + 7*B)*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(42*d) + (a*A*cos(c + d*x)^6*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 5, (3*(A - B)*arctanh(sin(c + d*x)))/(2*a*d) - (2*(3*A - 4*B)*tan(c + d*x))/(3*a*d) + (3*(A - B)*sec(c + d*x)*tan(c + d*x))/(2*a*d) - ((3*A - 4*B)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 4, -(((2*A - 3*B)*arctanh(sin(c + d*x)))/(2*a*d)) + (2*(A - B)*tan(c + d*x))/(a*d) - ((2*A - 3*B)*sec(c + d*x)*tan(c + d*x))/(2*a*d) + ((A - B)*sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 3, ((A - B)*arctanh(sin(c + d*x)))/(a*d) - ((A - 2*B)*tan(c + d*x))/(a*d) + ((A - B)*sec(c + d*x)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 2, (B*arctanh(sin(c + d*x)))/(a*d) + ((A - B)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 2, (A*x)/a - ((A - B)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 3, -(((A - B)*x)/a) + ((2*A - B)*sin(c + d*x))/(a*d) - ((A - B)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 4, ((3*A - 2*B)*x)/(2*a) - (2*(A - B)*sin(c + d*x))/(a*d) + ((3*A - 2*B)*cos(c + d*x)*sin(c + d*x))/(2*a*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x)), x, 5, -((3*(A - B)*x)/(2*a)) + (2*(4*A - 3*B)*sin(c + d*x))/(3*a*d) - (3*(A - B)*cos(c + d*x)*sin(c + d*x))/(2*a*d) + ((4*A - 3*B)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) - ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^5*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 6, ((7*A - 10*B)*arctanh(sin(c + d*x)))/(2*a^2*d) - (8*(2*A - 3*B)*tan(c + d*x))/(3*a^2*d) + ((7*A - 10*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - (4*(2*A - 3*B)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d) + ((A - B)*sec(c + d*x)^4*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((7*A - 10*B)*sec(c + d*x)^3*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 5, -(((4*A - 7*B)*arctanh(sin(c + d*x)))/(2*a^2*d)) + (2*(5*A - 8*B)*tan(c + d*x))/(3*a^2*d) - ((4*A - 7*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((5*A - 8*B)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 4, ((A - 2*B)*arctanh(sin(c + d*x)))/(a^2*d) - (2*(2*A - 5*B)*tan(c + d*x))/(3*a^2*d) + ((A - B)*sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A - 2*B)*sec(c + d*x)*tan(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 3, (B*arctanh(sin(c + d*x)))/(a^2*d) + ((A - B)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A - 4*B)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 2, ((A - B)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A + 2*B)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 3, (A*x)/a^2 - ((A - B)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((4*A - B)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 4, -(((2*A - B)*x)/a^2) + (2*(5*A - 2*B)*sin(c + d*x))/(3*a^2*d) - ((A - B)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((2*A - B)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 5, ((7*A - 4*B)*x)/(2*a^2) - (2*(8*A - 5*B)*sin(c + d*x))/(3*a^2*d) + ((7*A - 4*B)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((8*A - 5*B)*cos(c + d*x)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^2, x, 6, -(((10*A - 7*B)*x)/(2*a^2)) + (8*(3*A - 2*B)*sin(c + d*x))/(3*a^2*d) - ((10*A - 7*B)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + (4*(3*A - 2*B)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) - ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((10*A - 7*B)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^5*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 6, -(((6*A - 13*B)*arctanh(sin(c + d*x)))/(2*a^3*d)) + (8*(9*A - 19*B)*tan(c + d*x))/(15*a^3*d) - ((6*A - 13*B)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) + ((A - B)*sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((6*A - 11*B)*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (4*(9*A - 19*B)*sec(c + d*x)^2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 5, ((A - 3*B)*arctanh(sin(c + d*x)))/(a^3*d) - (2*(11*A - 36*B)*tan(c + d*x))/(15*a^3*d) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((4*A - 9*B)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A - 3*B)*sec(c + d*x)*tan(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 4, (B*arctanh(sin(c + d*x)))/(a^3*d) + ((A - B)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((2*A - 7*B)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*(A - 11*B)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 3, -(((A - B)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3)) - ((3*A + 2*B)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*(3*A + 2*B)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 3, ((A - B)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((2*A + 3*B)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((2*A + 3*B)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 4, (A*x)/a^3 - ((A - B)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((7*A - 2*B)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (2*(11*A - B)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 5, -(((3*A - B)*x)/a^3) + (2*(36*A - 11*B)*sin(c + d*x))/(15*a^3*d) - ((A - B)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((9*A - 4*B)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((3*A - B)*sin(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 6, ((13*A - 6*B)*x)/(2*a^3) - (8*(19*A - 9*B)*sin(c + d*x))/(15*a^3*d) + ((13*A - 6*B)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((11*A - 6*B)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (4*(19*A - 9*B)*cos(c + d*x)*sin(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^3, x, 7, -(((23*A - 13*B)*x)/(2*a^3)) + (8*(34*A - 19*B)*sin(c + d*x))/(15*a^3*d) - ((23*A - 13*B)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (4*(34*A - 19*B)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) - ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((13*A - 8*B)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((23*A - 13*B)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^6*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 7, -(((8*A - 21*B)*arctanh(sin(c + d*x)))/(2*a^4*d)) + (8*(83*A - 216*B)*tan(c + d*x))/(105*a^4*d) - ((8*A - 21*B)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) + ((A - B)*sec(c + d*x)^5*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((A - 2*B)*sec(c + d*x)^4*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) + ((52*A - 129*B)*sec(c + d*x)^3*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (4*(83*A - 216*B)*sec(c + d*x)^2*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^5*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 6, ((A - 4*B)*arctanh(sin(c + d*x)))/(a^4*d) - (8*(20*A - 83*B)*tan(c + d*x))/(105*a^4*d) + ((A - B)*sec(c + d*x)^4*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((5*A - 12*B)*sec(c + d*x)^3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((25*A - 88*B)*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + ((A - 4*B)*sec(c + d*x)*tan(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 5, (B*arctanh(sin(c + d*x)))/(a^4*d) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((3*A - 10*B)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((6*A - 55*B)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (2*(3*A - 80*B)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 4, -(((A - B)*sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) + ((4*A + 3*B)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - (2*(4*A + 3*B)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (4*(4*A + 3*B)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 4, ((A - B)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((A + 6*B)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((8*A + 13*B)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + ((8*A + 13*B)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 4, ((A - B)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((3*A + 4*B)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + (2*(3*A + 4*B)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (2*(3*A + 4*B)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 5, (A*x)/a^4 - ((A - B)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((10*A - 3*B)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((55*A - 6*B)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (2*(80*A - 3*B)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 6, -(((4*A - B)*x)/a^4) + (8*(83*A - 20*B)*sin(c + d*x))/(105*a^4*d) - ((A - B)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((12*A - 5*B)*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((88*A - 25*B)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - ((4*A - B)*sin(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 7, ((21*A - 8*B)*x)/(2*a^4) - (8*(216*A - 83*B)*sin(c + d*x))/(105*a^4*d) + ((21*A - 8*B)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((2*A - B)*cos(c + d*x)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - ((129*A - 52*B)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*(216*A - 83*B)*cos(c + d*x)*sin(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^4, x, 8, -(((44*A - 21*B)*x)/(2*a^4)) + (16*(227*A - 108*B)*sin(c + d*x))/(105*a^4*d) - ((44*A - 21*B)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) + (8*(227*A - 108*B)*cos(c + d*x)^2*sin(c + d*x))/(105*a^4*d) - ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((16*A - 9*B)*cos(c + d*x)^2*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((178*A - 87*B)*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - ((44*A - 21*B)*cos(c + d*x)^2*sin(c + d*x))/(3*a^3*d*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 5, (16*a*(9*A + 8*B)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (4*a*(9*A + 8*B)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(9*A + 8*B)*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a*B*sec(c + d*x)^4*tan(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x))) + (16*(9*A + 8*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d)],
[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 4, (8*a*(7*A + 6*B)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(7*A + 6*B)*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*B*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x))) + (8*(7*A + 6*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 3, (2*a*(5*A + 4*B)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*B*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*(5*A + 4*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 2, (2*a*(3*A + B)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*B*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 3, (2*sqrt(a)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a*B*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 2, (sqrt(a)*(A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*A*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 3, (sqrt(a)*(3*A + 4*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a*(3*A + 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 4, (sqrt(a)*(5*A + 6*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a*(5*A + 6*B)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*(5*A + 6*B)*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/2), x, 5, (5*sqrt(a)*(7*A + 8*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (5*a*(7*A + 8*B)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (5*a*(7*A + 8*B)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*(7*A + 8*B)*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 5, (8*a^2*(39*A + 34*B)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(39*A + 34*B)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(9*A + 10*B)*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (8*a*(39*A + 34*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a*B*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 4, (2*a^2*(63*A + 52*B)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(7*A + 8*B)*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*(63*A + 52*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*a*B*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 3, (8*a^2*(5*A + 3*B)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*(5*A + 3*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*B*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 4, (2*a^(3/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*(3*A + 4*B)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*B*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 3, (a^(3/2)*(3*A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*(A - 2*B)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (2*a*B*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 3, (a^(3/2)*(7*A + 12*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^2*(5*A + 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 4, (a^(3/2)*(11*A + 14*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^2*(11*A + 14*B)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a^2*(7*A + 6*B)*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(3/2), x, 5, (a^(3/2)*(75*A + 88*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^2*(75*A + 88*B)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(75*A + 88*B)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a^2*(9*A + 8*B)*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 6, (8*a^3*(803*A + 710*B)*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(803*A + 710*B)*sec(c + d*x)^2*tan(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(209*A + 194*B)*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (8*a^2*(803*A + 710*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3465*d) + (2*a^2*(11*A + 14*B)*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(99*d) + (2*a*B*sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 5, (2*a^3*(345*A + 292*B)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(135*A + 124*B)*sec(c + d*x)^2*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(345*A + 292*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a^2*(3*A + 4*B)*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*B*sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (64*a^3*(7*A + 5*B)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(7*A + 5*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*a*(7*A + 5*B)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*B*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 5, (2*a^(5/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^3*(35*A + 32*B)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(5*A + 8*B)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*a*B*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (a^(5/2)*(5*A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d - (a^3*(3*A + 14*B)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(A + 2*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d + (2*a*B*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (a^(5/2)*(19*A + 20*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^3*(9*A - 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) - (a^2*(A - 4*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 4, (a^(5/2)*(25*A + 38*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^3*(49*A + 54*B)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*(3*A + 2*B)*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 5, (a^(5/2)*(163*A + 200*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^3*(163*A + 200*B)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^3*(95*A + 104*B)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a^2*(11*A + 8*B)*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (a*A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(5/2), x, 6, (a^(5/2)*(283*A + 326*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(128*d) + (a^3*(283*A + 326*B)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(283*A + 326*B)*cos(c + d*x)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^3*(157*A + 170*B)*cos(c + d*x)^2*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (a^2*(13*A + 10*B)*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 5, -((sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(91*A - 43*B)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*(7*A - 31*B)*sec(c + d*x)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*(7*A - B)*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*B*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 4, (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*(5*A - 13*B)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*(5*A - B)*sec(c + d*x)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*B*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 3, -((sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(3*A - B)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*B*sec(c + d*x)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 2, (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*B*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 3, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 4, -(((A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d)) + (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (A*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 5, ((7*A - 4*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - ((A - 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/2), x, 6, -(((9*A - 14*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*sqrt(a)*d)) + (sqrt(2)*(A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + ((7*A - 2*B)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) - ((A - 6*B)*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 5, ((11*A - 15*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((95*A - 147*B)*tan(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) + ((35*A - 39*B)*sec(c + d*x)*tan(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) - ((5*A - 9*B)*sec(c + d*x)^2*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 4, -(((7*A - 11*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) + ((A - B)*sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((15*A - 19*B)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x))) - ((3*A - 7*B)*sec(c + d*x)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 3, ((3*A - 7*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((A - 5*B)*tan(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 2, ((A + 3*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - ((5*A - B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 5, -(((3*A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d)) + ((9*A - 5*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((3*A - B)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 6, ((19*A - 12*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(3/2)*d) - ((13*A - 9*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((7*A - 6*B)*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + ((2*A - B)*cos(c + d*x)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 7, -(((47*A - 38*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*a^(3/2)*d)) + ((17*A - 13*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (7*(3*A - 2*B)*sin(c + d*x))/(8*a*d*sqrt(a + a*sec(c + d*x))) - ((13*A - 12*B)*cos(c + d*x)*sin(c + d*x))/(12*a*d*sqrt(a + a*sec(c + d*x))) + ((5*A - 3*B)*cos(c + d*x)^2*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 5, -(((75*A - 163*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) + ((A - B)*sec(c + d*x)^3*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((9*A - 17*B)*sec(c + d*x)^2*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((147*A - 299*B)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x))) - ((39*A - 95*B)*sec(c + d*x)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 4, ((19*A - 75*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((5*A - 13*B)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((9*A - 49*B)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 3, ((5*A + 19*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((A - 9*B)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 3, ((3*A + 5*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((3*A + 5*B)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 5, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - ((43*A - 3*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((11*A - 3*B)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 6, -(((5*A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d)) + ((115*A - 43*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((15*A - 7*B)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((35*A - 11*B)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 7, ((39*A - 20*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(5/2)*d) - ((219*A - 115*B)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*cos(c + d*x)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((19*A - 11*B)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - (7*(9*A - 5*B)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + ((31*A - 15*B)*cos(c + d*x)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+A Sec[c+d x]) (a - a Sec[c+d x])^(n/2)


[sec(c + d*x)^0*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(1/2), x, 3, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(sqrt(a)*d) - (2*sqrt(2)*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^1*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(1/2), x, 4, (3*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(sqrt(a)*d) - (2*sqrt(2)*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d) + (A*sin(c + d*x))/(d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(1/2), x, 5, (11*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(4*sqrt(a)*d) - (2*sqrt(2)*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d) + (5*A*sin(c + d*x))/(4*d*sqrt(a - a*sec(c + d*x))) + (A*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(1/2), x, 6, (23*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(8*sqrt(a)*d) - (2*sqrt(2)*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(a)*d) + (9*A*sin(c + d*x))/(8*d*sqrt(a - a*sec(c + d*x))) + (7*A*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a - a*sec(c + d*x))) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a - a*sec(c + d*x)))],


[sec(c + d*x)^0*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(3/2), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(a^(3/2)*d) - (3*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(2)*a^(3/2)*d) - (A*tan(c + d*x))/(d*(a - a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(3/2), x, 5, (5*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(a^(3/2)*d) - (7*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(2)*a^(3/2)*d) - (A*sin(c + d*x))/(d*(a - a*sec(c + d*x))^(3/2)) + (2*A*sin(c + d*x))/(a*d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(3/2), x, 6, (31*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(4*a^(3/2)*d) - (11*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(2)*a^(3/2)*d) - (A*cos(c + d*x)*sin(c + d*x))/(d*(a - a*sec(c + d*x))^(3/2)) + (13*A*sin(c + d*x))/(4*a*d*sqrt(a - a*sec(c + d*x))) + (3*A*cos(c + d*x)*sin(c + d*x))/(2*a*d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(3/2), x, 7, (85*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(8*a^(3/2)*d) - (15*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(sqrt(2)*a^(3/2)*d) - (A*cos(c + d*x)^2*sin(c + d*x))/(d*(a - a*sec(c + d*x))^(3/2)) + (35*A*sin(c + d*x))/(8*a*d*sqrt(a - a*sec(c + d*x))) + (25*A*cos(c + d*x)*sin(c + d*x))/(12*a*d*sqrt(a - a*sec(c + d*x))) + (4*A*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*sqrt(a - a*sec(c + d*x)))],


[sec(c + d*x)^0*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(5/2), x, 5, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(a^(5/2)*d) - (23*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(8*sqrt(2)*a^(5/2)*d) - (A*tan(c + d*x))/(2*d*(a - a*sec(c + d*x))^(5/2)) - (7*A*tan(c + d*x))/(8*a*d*(a - a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(5/2), x, 6, (7*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(a^(5/2)*d) - (79*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(8*sqrt(2)*a^(5/2)*d) - (A*sin(c + d*x))/(2*d*(a - a*sec(c + d*x))^(5/2)) - (11*A*sin(c + d*x))/(8*a*d*(a - a*sec(c + d*x))^(3/2)) + (23*A*sin(c + d*x))/(8*a^2*d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(5/2), x, 7, (59*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(4*a^(5/2)*d) - (167*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(8*sqrt(2)*a^(5/2)*d) - (A*cos(c + d*x)*sin(c + d*x))/(2*d*(a - a*sec(c + d*x))^(5/2)) - (15*A*cos(c + d*x)*sin(c + d*x))/(8*a*d*(a - a*sec(c + d*x))^(3/2)) + (49*A*sin(c + d*x))/(8*a^2*d*sqrt(a - a*sec(c + d*x))) + (23*A*cos(c + d*x)*sin(c + d*x))/(8*a^2*d*sqrt(a - a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + A*sec(c + d*x))/(a - a*sec(c + d*x))^(5/2), x, 8, (203*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a - a*sec(c + d*x))))/(8*a^(5/2)*d) - (287*A*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a - a*sec(c + d*x)))))/(8*sqrt(2)*a^(5/2)*d) - (A*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a - a*sec(c + d*x))^(5/2)) - (19*A*cos(c + d*x)^2*sin(c + d*x))/(8*a*d*(a - a*sec(c + d*x))^(3/2)) + (21*A*sin(c + d*x))/(2*a^2*d*sqrt(a - a*sec(c + d*x))) + (119*A*cos(c + d*x)*sin(c + d*x))/(24*a^2*d*sqrt(a - a*sec(c + d*x))) + (77*A*cos(c + d*x)^2*sin(c + d*x))/(24*a^2*d*sqrt(a - a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[sec(c + d*x)^(5/2)*(A + B*sec(c + d*x)), x, 7, (-6*B*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*A*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (6*B*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*B*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x)), x, 6, (-2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sqrt(sec(c + d*x))*(A + B*sec(c + d*x)), x, 5, (-2*B*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*A*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(A + B*sec(c + d*x))/sqrt(sec(c + d*x)), x, 4, (2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d],
[(A + B*sec(c + d*x))/sec(c + d*x)^(3/2), x, 5, (2*B*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*A*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x))/sec(c + d*x)^(5/2), x, 6, (6*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*B*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x))/sec(c + d*x)^(7/2), x, 7, (6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (10*A*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*A*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 8, (-6*a*(A + B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(7*A + 5*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (6*a*(A + B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(7*A + 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*(A + B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a*B*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 7, (-2*a*(5*A + 3*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(5*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(A + B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a*B*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 6, (-2*a*(A + B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(3*A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(A + B)*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sqrt(sec(c + d*x)), x, 5, (2*a*(A - B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*B*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(3/2), x, 5, (2*a*(A + B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(A + 3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(5/2), x, 6, (2*a*(3*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a*(A + B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(7/2), x, 7, (6*a*(A + B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(5*A + 7*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*a*(A + B)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a*(5*A + 7*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 8, -((4*a^2*(4*A + 3*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^2*(7*A + 6*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(4*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^2*(7*A + 6*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a^2*(7*A + 9*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*B*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 7, -((4*a^2*(5*A + 4*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^2*(2*A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^2*(5*A + 4*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(5*A + 7*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*B*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sqrt(sec(c + d*x)), x, 6, -((4*a^2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (4*a^2*(3*A + 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(3*A + 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*B*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sec(c + d*x)^(3/2), x, 6, (4*a^2*A*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (4*a^2*(2*A + 3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sec(c + d*x)^(5/2), x, 6, (4*a^2*(4*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(A + 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(7*A + 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sec(c + d*x)^(7/2), x, 7, (4*a^2*(3*A + 4*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(6*A + 7*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(9*A + 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (4*a^2*(6*A + 7*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sec(c + d*x)^(9/2), x, 8, (4*a^2*(8*A + 9*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(5*A + 6*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(11*A + 9*B)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (4*a^2*(8*A + 9*B)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (4*a^2*(5*A + 6*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 9, -((4*a^3*(21*A + 17*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (4*a^3*(13*A + 11*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(21*A + 17*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(13*A + 11*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*a^3*(24*A + 23*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (2*a^2*(9*A + 13*B)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*a*B*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 8, -((4*a^3*(9*A + 7*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*(21*A + 13*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(9*A + 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^3*(42*A + 41*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 11*B)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a*B*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sqrt(sec(c + d*x)), x, 7, -((4*a^3*(5*A + 9*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*(5*A + 3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*(20*A + 21*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*(5*A + 9*B)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*B*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sec(c + d*x)^(3/2), x, 7, (4*a^3*(A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (20*a^3*(A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*(A + 4*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) - (2*a^2*(A - B)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sec(c + d*x)^(5/2), x, 7, (4*a^3*(9*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(3*A + 5*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(6*A - 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*(9*A + 5*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sec(c + d*x)^(7/2), x, 7, (4*a^3*(7*A + 9*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(13*A + 21*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(41*A + 42*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(11*A + 7*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sec(c + d*x)^(9/2), x, 8, (4*a^3*(17*A + 21*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(11*A + 13*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(23*A + 24*B)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (4*a^3*(11*A + 13*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a^2*(13*A + 9*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sec(c + d*x)^(11/2), x, 9, (4*a^3*(15*A + 17*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(105*A + 121*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (20*a^3*(21*A + 22*B)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (4*a^3*(15*A + 17*B)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (4*a^3*(105*A + 121*B)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(15*A + 11*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


# ::Subsubsection::Closed:: 
#n<0


[(sec(c + d*x)^(7/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 8, (3*(5*A - 7*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d) + (5*(A - B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) - (3*(5*A - 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d) + (5*(A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - ((5*A - 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) + ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 7, -((3*(A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - ((3*A - 5*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (3*(A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - ((3*A - 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) + ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 6, ((A - 3*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + ((A - B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 5, -(((A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + ((A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))), x, 5, ((3*A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, -((3*(A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + ((5*A - 3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((5*A - 3*B)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (3*(7*A - 5*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d) - (5*(A - B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((7*A - 5*B)*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - (5*(A - B)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))), x, 8, -((21*(A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d)) + (5*(9*A - 7*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*a*d) + ((9*A - 7*B)*sin(c + d*x))/(7*a*d*sec(c + d*x)^(5/2)) - (7*(A - B)*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) + (5*(9*A - 7*B)*sin(c + d*x))/(21*a*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(d*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(7/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 8, -(((4*A - 7*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) - (5*(A - 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((4*A - 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - (5*(A - 2*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) + ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((4*A - 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 7, ((A - 4*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((2*A - 5*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - 4*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((2*A - 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 6, (B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((A + 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (B*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 6, -((A*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + ((2*A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((2*A + B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2), x, 6, ((4*A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) - ((5*A - 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((5*A - 2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 7, -(((7*A - 4*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (5*(2*A - B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (5*(2*A - B)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - ((7*A - 4*B)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 8, (7*(8*A - 5*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*(3*A - 2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (7*(8*A - 5*B)*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*(3*A - 2*B)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((3*A - 2*B)*sin(c + d*x))/(a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(9/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 9, -((7*(7*A - 17*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) - ((13*A - 33*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (7*(7*A - 17*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((13*A - 33*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^3*d) + ((A - B)*sec(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((A - 2*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) + (7*(7*A - 17*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(7/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 8, ((9*A - 49*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A - 13*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((9*A - 49*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((3*A - 8*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A - 13*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 7, ((A + 9*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((A - 6*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((A + 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 7, -(((A - B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + ((A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((A + 4*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A + B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 7, -(((9*A + B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + ((3*A + B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((3*A + 2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A + B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3), x, 7, ((49*A - 9*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((8*A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((13*A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 8, -((7*(17*A - 7*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + ((33*A - 13*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((33*A - 13*B)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - ((2*A - B)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - (7*(17*A - 7*B)*sin(c + d*x))/(30*a^2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 9, (7*(33*A - 17*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((21*A - 11*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (7*(33*A - 17*B)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) - ((21*A - 11*B)*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - ((12*A - 7*B)*sin(c + d*x))/(15*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (3*(21*A - 11*B)*sin(c + d*x))/(10*a^2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]) (a+a Sec[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 5, (a*(6*A + 5*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(6*A + 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*(6*A + 5*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 4, (a*(4*A + 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(4*A + 3*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 3, (sqrt(2)*a*(2*A + B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sqrt(sec(c + d*x)), x, 3, (2*sqrt(2)*a*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(3/2), x, 2, (2*a*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(2*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(5/2), x, 3, (2*a*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(4*A + 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a*(4*A + 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sec(c + d*x)^(7/2), x, 4, (2*a*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(6*A + 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a*(6*A + 7*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*(6*A + 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 6, (a^2*(88*A + 75*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(88*A + 75*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(88*A + 75*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 5, (a^2*(14*A + 11*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(14*A + 11*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a^2*(6*A + 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 4, (a^2*(12*A + 7*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(4*A + 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*B*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x))/sqrt(sec(c + d*x)), x, 4, (sqrt(2)*a^2*(2*A + 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*(2*A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (a*B*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(3/2), x, 4, (2*sqrt(2)*a^2*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(5/2), x, 3, (8*a^2*(3*A + 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(7/2), x, 4, (2*a^2*(8*A + 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(52*A + 63*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^2*(52*A + 63*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(9/2), x, 5, (2*a^2*(10*A + 9*B)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(34*A + 39*B)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^2*(34*A + 39*B)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(34*A + 39*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 7, (a^3*(326*A + 283*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(326*A + 283*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(326*A + 283*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^3*(170*A + 157*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (a^2*(10*A + 13*B)*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (a*B*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 6, (a^3*(200*A + 163*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(200*A + 163*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^3*(104*A + 95*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 11*B)*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (a*B*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 5, (a^3*(38*A + 25*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(54*A + 49*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*(2*A + 3*B)*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*B*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sqrt(sec(c + d*x)), x, 5, (a^3*(20*A + 19*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(4*A - 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a^2*(4*A + 7*B)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*B*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(3/2), x, 5, (sqrt(2)*a^3*(2*A + 5*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^3*(14*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) - (a^2*(2*A - 3*B)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(5/2), x, 5, (2*sqrt(2)*a^3*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(32*A + 35*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(8*A + 5*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(7/2), x, 4, (64*a^3*(5*A + 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(5*A + 7*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a*(5*A + 7*B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(9/2), x, 5, (2*a^3*(124*A + 135*B)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(292*A + 345*B)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^3*(292*A + 345*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 3*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x))/sec(c + d*x)^(11/2), x, 6, (2*a^3*(194*A + 209*B)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(710*A + 803*B)*sin(c + d*x))/(1155*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^3*(710*A + 803*B)*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^3*(710*A + 803*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(14*A + 11*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(5/2)*(A + B*sec(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 7, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) - ((4*A - 7*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + ((4*A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (B*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*A - B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (B*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[sqrt(sec(c + d*x))*(A + B*sec(c + d*x))/sqrt(a + a*sec(c + d*x)), x, 5, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 3, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 4, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 5, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (2*(A - 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(13*A - 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - (2*(A - 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*(31*A - 7*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A - 91*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(7/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 8, ((9*A - 13*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((12*A - 19*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((6*A - 7*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) - ((A - 2*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 7, -((5*A - 9*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*A - 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((A - 3*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 6, ((A - 5*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[sqrt(sec(c + d*x))*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(3/2), x, 3, ((3*A + B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 4, -((7*A - 3*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((5*A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 5, ((11*A - 7*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((7*A - 3*B)*sin(c + d*x))/(6*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((19*A - 15*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, -((15*A - 11*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((9*A - 5*B)*sin(c + d*x))/(10*a*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((39*A - 35*B)*sin(c + d*x))/(30*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((147*A - 95*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^(7/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 8, -(((43*A - 115*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))) + (sqrt(2)*(2*A - 5*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((7*A - 15*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((11*A - 35*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^(5/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 7, ((3*A - 43*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((3*A - 11*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 4, ((5*A + 3*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sqrt(sec(c + d*x))*(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(5/2), x, 5, ((19*A + 5*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((9*A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)), ((19*A + 5*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((9*A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, -((75*A - 19*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((49*A - 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 6, ((163*A - 75*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((17*A - 9*B)*sin(c + d*x))/(16*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((95*A - 39*B)*sin(c + d*x))/(48*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((299*A - 147*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, -((283*A - 163*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - ((21*A - 13*B)*sin(c + d*x))/(16*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((157*A - 85*B)*sin(c + d*x))/(80*a^2*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((787*A - 475*B)*sin(c + d*x))/(240*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((2671*A - 1495*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (a+a Sec[c+d x])^(n/3)
#


[(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(2/3), x, 1, Int((a + a*sec(c + d*x))^(2/3)*(A + B*sec(c + d*x)), x)],
[(A + B*sec(c + d*x))*(a + a*sec(c + d*x))^(1/3), x, 1, Int((a + a*sec(c + d*x))^(1/3)*(A + B*sec(c + d*x)), x)],
[(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/3), x, 0, Int((A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(1/3), x)],
[(A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(2/3), x, 0, Int((A + B*sec(c + d*x))/(a + a*sec(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Sec[c+d x]) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(A + B*sec(c + d*x))*cos(c + d*x)^(7/2), x, 6, (6*B*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (10*A*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (10*A*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[(A + B*sec(c + d*x))*cos(c + d*x)^(5/2), x, 5, (6*A*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[(A + B*sec(c + d*x))*cos(c + d*x)^(3/2), x, 4, (2*B*EllipticE((1/2)*(c + d*x), 2))/d + (2*A*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[(A + B*sec(c + d*x))*cos(c + d*x)^(1/2), x, 3, (2*A*EllipticE((1/2)*(c + d*x), 2))/d + (2*B*EllipticF((1/2)*(c + d*x), 2))/d],
[(A + B*sec(c + d*x))/cos(c + d*x)^(1/2), x, 4, -((2*B*EllipticE((1/2)*(c + d*x), 2))/d) + (2*A*EllipticF((1/2)*(c + d*x), 2))/d + (2*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(A + B*sec(c + d*x))/cos(c + d*x)^(3/2), x, 5, -((2*A*EllipticE((1/2)*(c + d*x), 2))/d) + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(A + B*sec(c + d*x))/cos(c + d*x)^(5/2), x, 6, -((6*B*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*A*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*B*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 6, (6*a*(A + B)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(5*A + 7*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(5*A + 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(A + B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 5, (2*a*(3*A + 5*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(A + B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(A + B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 4, (2*a*(A + B)*EllipticE((c + d*x)/2, 2))/d + (2*a*(A + 3*B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 4, (2*a*(A - B)*EllipticE((c + d*x)/2, 2))/d + (2*a*(A + B)*EllipticF((c + d*x)/2, 2))/d + (2*a*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 5, (-2*a*(A + B)*EllipticE((c + d*x)/2, 2))/d + (2*a*(3*A + B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*(A + B)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 6, (-2*a*(5*A + 3*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(A + B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*B*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a*(A + B)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 3*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 7, (4*a^2*(8*A + 9*B)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(5*A + 6*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^2*(5*A + 6*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a^2*(8*A + 9*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(11*A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 6, (4*a^2*(3*A + 4*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(6*A + 7*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^2*(6*A + 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(9*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*a*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 5, (4*a^2*(4*A + 5*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(A + 2*B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(7*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 5, (4*a^2*A*EllipticE((c + d*x)/2, 2))/d + (4*a^2*(2*A + 3*B)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(A - 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)), x, 5, (-4*a^2*B*EllipticE((c + d*x)/2, 2))/d + (4*a^2*(3*A + 2*B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(3*A + 5*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*B*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 6, (-4*a^2*(5*A + 4*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(2*A + B)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(5*A + 7*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (4*a^2*(5*A + 4*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*B*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 7, (-4*a^2*(4*A + 3*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(7*A + 9*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (4*a^2*(7*A + 6*B)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^2*(4*A + 3*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*B*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 8, (4*a^3*(15*A + 17*B)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(105*A + 121*B)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(105*A + 121*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(15*A + 17*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (20*a^3*(21*A + 22*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*a^2*(15*A + 11*B)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*a*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 7, (4*a^3*(17*A + 21*B)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(11*A + 13*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(11*A + 13*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a^3*(23*A + 24*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(13*A + 9*B)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 6, (4*a^3*(7*A + 9*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(13*A + 21*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(41*A + 42*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(11*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 6, (4*a^3*(9*A + 5*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(3*A + 5*B)*EllipticF((c + d*x)/2, 2))/(3*d) - (4*a^3*(6*A - 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(9*A + 5*B)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 6, (4*a^3*(A - B)*EllipticE((c + d*x)/2, 2))/d + (20*a^3*(A + B)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a^3*(A + 4*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) - (2*a^2*(A - B)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)), x, 6, (-4*a^3*(5*A + 9*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(5*A + 3*B)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a^3*(20*A + 21*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(5*A + 9*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*B*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 7, (-4*a^3*(9*A + 7*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(21*A + 13*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(42*A + 41*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(9*A + 7*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(7*A + 11*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 8, (-4*a^3*(21*A + 17*B)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(13*A + 11*B)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(24*A + 23*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^3*(13*A + 11*B)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^3*(21*A + 17*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(9*A + 13*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(5/2)) + (2*a*B*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(5/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 6, (3*(7*A - 5*B)*EllipticE((c + d*x)/2, 2))/(5*a*d) - (5*(A - B)*EllipticF((c + d*x)/2, 2))/(3*a*d) - (5*(A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + ((7*A - 5*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 5, (-3*(A - B)*EllipticE((c + d*x)/2, 2))/(a*d) + ((5*A - 3*B)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((5*A - 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x)), x, 4, ((3*A - B)*EllipticE((c + d*x)/2, 2))/(a*d) - ((A - B)*EllipticF((c + d*x)/2, 2))/(a*d) - ((A - B)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))), x, 4, -(((A - B)*EllipticE((c + d*x)/2, 2))/(a*d)) + ((A + B)*EllipticF((c + d*x)/2, 2))/(a*d) + ((A - B)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 5, ((A - 3*B)*EllipticE((c + d*x)/2, 2))/(a*d) + ((A - B)*EllipticF((c + d*x)/2, 2))/(a*d) - ((A - 3*B)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 6, (-3*(A - B)*EllipticE((c + d*x)/2, 2))/(a*d) - ((3*A - 5*B)*EllipticF((c + d*x)/2, 2))/(3*a*d) - ((3*A - 5*B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) + (3*(A - B)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 7, (7*(8*A - 5*B)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(3*A - 2*B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (5*(3*A - 2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + (7*(8*A - 5*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((3*A - 2*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 6, -(((7*A - 4*B)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + (5*(2*A - B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (5*(2*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((7*A - 4*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^2, x, 5, ((4*A - B)*EllipticE((c + d*x)/2, 2))/(a^2*d) - ((5*A - 2*B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A - B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((5*A - 2*B)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2), x, 5, -((A*EllipticE((c + d*x)/2, 2))/(a^2*d)) + ((2*A + B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + ((A - B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((2*A + B)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 5, (B*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((A + 2*B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + ((A - B)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - (B*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 6, ((A - 4*B)*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((2*A - 5*B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A - 4*B)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((2*A - 5*B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2), x, 7, -(((4*A - 7*B)*EllipticE((c + d*x)/2, 2))/(a^2*d)) - (5*(A - 2*B)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (5*(A - 2*B)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + ((4*A - 7*B)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) + ((4*A - 7*B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 8, (7*(33*A - 17*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - ((21*A - 11*B)*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) - ((21*A - 11*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) + (7*(33*A - 17*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((12*A - 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (3*(21*A - 11*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 7, (-7*(17*A - 7*B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((33*A - 13*B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((33*A - 13*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((2*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - (7*(17*A - 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^3, x, 6, ((49*A - 9*B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((13*A - 3*B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A - B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - ((8*A - 3*B)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((13*A - 3*B)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3), x, 6, -((9*A + B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A + B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) + ((3*A + 2*B)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((3*A + B)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 6, -((A - B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - ((A + 4*B)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((A + B)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 6, ((A + 9*B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 3*B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) + ((A - 6*B)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((A + 9*B)*sin(c + d*x))/(10*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3), x, 7, ((9*A - 49*B)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A - 13*B)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((9*A - 49*B)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) + ((3*A - 8*B)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((3*A - 13*B)*sin(c + d*x))/(6*a^2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3), x, 8, -((7*(7*A - 17*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) - ((13*A - 33*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((13*A - 33*B)*sin(c + d*x))/(6*a^3*d*cos(c + d*x)^(3/2)) + (7*(7*A - 17*B)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) + ((A - B)*sin(c + d*x))/(5*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) + ((A - 2*B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) + (7*(7*A - 17*B)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Sec[c+d x]) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(9/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 5, (32*a*(8*A + 9*B)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*(8*A + 9*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (4*a*(8*A + 9*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(8*A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 4, (16*a*(6*A + 7*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a*(6*A + 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(6*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 3, (4*a*(4*A + 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(4*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 2, (2*a*(2*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)), x, 3, (2*sqrt(2)*a*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 3, (sqrt(2)*a*(2*A + B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*B*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 4, (a*(4*A + 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*B*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(4*A + 3*B)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x)))/cos(c + d*x)^(5/2), x, 5, (a*(6*A + 5*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*B*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a*(6*A + 5*B)*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(6*A + 5*B)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 6, (32*a^2*(168*A + 187*B)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(168*A + 187*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (4*a^2*(168*A + 187*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(168*A + 187*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(12*A + 11*B)*cos(c + d*x)^(7/2)*sin(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(9/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 5, (16*a^2*(34*A + 39*B)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^2*(34*A + 39*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(34*A + 39*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(10*A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 4, (4*a^2*(52*A + 63*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(52*A + 63*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(8*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 3, (8*a^2*(3*A + 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 4, (2*sqrt(2)*a^2*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 4, (sqrt(2)*a^2*(2*A + 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(2*A - B)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*B*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 4, (a^2*(12*A + 7*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(4*A + 5*B)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*B*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 5, (a^2*(14*A + 11*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(6*A + 7*B)*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(14*A + 11*B)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*B*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(5/2), x, 6, (a^2*(88*A + 75*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 9*B)*sin(c + d*x))/(24*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(88*A + 75*B)*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(88*A + 75*B)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*B*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 6, (16*a^3*(710*A + 803*B)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^3*(710*A + 803*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(710*A + 803*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(194*A + 209*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(14*A + 11*B)*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*a*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 5, (4*a^3*(292*A + 345*B)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(292*A + 345*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(124*A + 135*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 3*B)*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 4, (64*a^3*(5*A + 7*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(5*A + 7*B)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(5*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 5, (2*sqrt(2)*a^3*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(32*A + 35*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(8*A + 5*B)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 5, (sqrt(2)*a^3*(2*A + 5*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(14*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a^2*(2*A - 3*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 5, (a^3*(20*A + 19*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(4*A - 9*B)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(4*A + 7*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (a*B*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sqrt(cos(c + d*x)), x, 5, (a^3*(38*A + 25*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(54*A + 49*B)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(2*A + 3*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)) + (a*B*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 6, (a^3*(200*A + 163*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(104*A + 95*B)*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(200*A + 163*B)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 11*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d*cos(c + d*x)^(5/2)) + (a*B*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(5/2), x, 7, (a^3*(326*A + 283*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(170*A + 157*B)*sin(c + d*x))/(240*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(326*A + 283*B)*sin(c + d*x))/(192*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(326*A + 283*B)*sin(c + d*x))/(128*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(10*A + 13*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d*cos(c + d*x)^(7/2)) + (a*B*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + B*sec(c + d*x)))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A - 91*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(31*A - 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*(A - 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/sqrt(a + a*sec(c + d*x)), x, 5, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(13*A - 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/sqrt(a + a*sec(c + d*x)), x, 4, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/sqrt(a + a*sec(c + d*x)), x, 3, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 5, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 6, (-2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*A - B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (B*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 7, (2*(A - B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((4*A - 7*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (B*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + ((4*A - B)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],

[(sqrt(cos(c + d*x))*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(a*A - A*b - a*B + b*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*b*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(3/2), x, 6, -((15*A - 11*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((147*A - 95*B)*sin(c + d*x))/(30*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((39*A - 35*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) + ((9*A - 5*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(3/2), x, 5, ((11*A - 7*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((19*A - 15*B)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((7*A - 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(3/2), x, 4, -((7*A - 3*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((5*A - B)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 3, ((3*A + B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, ((A - 5*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 7, -((5*A - 9*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*A - 3*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) - ((A - 3*B)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)), x, 8, ((9*A - 13*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((12*A - 19*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(2*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)) - ((A - 2*B)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + ((6*A - 7*B)*sin(c + d*x))/(4*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(5/2), x, 7, -((283*A - 163*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((21*A - 13*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((2671*A - 1495*B)*sin(c + d*x))/(240*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((787*A - 475*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) + ((157*A - 85*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(5/2), x, 6, ((163*A - 75*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((17*A - 9*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((299*A - 147*B)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((95*A - 39*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x)))/(a + a*sec(c + d*x))^(5/2), x, 5, -((75*A - 19*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 5*B)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((49*A - 9*B)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, ((19*A + 5*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) - ((9*A - B)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 4, ((5*A + 3*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, ((3*A - 43*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + ((3*A - 11*B)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)), x, 8, -(((43*A - 115*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))) + (sqrt(2)*(2*A - 5*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((A - B)*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)) + ((7*A - 15*B)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) - ((11*A - 35*B)*sin(c + d*x))/(16*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


# ::Section:: 
#Sec[c+d x]^m (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>=0


[sec(c + d*x)^6*(A + C*sec(c + d*x)^2), x, 3, ((7*A + 6*C)*tan(c + d*x))/(7*d) + (C*sec(c + d*x)^6*tan(c + d*x))/(7*d) + (2*(7*A + 6*C)*tan(c + d*x)^3)/(21*d) + ((7*A + 6*C)*tan(c + d*x)^5)/(35*d)],
[sec(c + d*x)^5*(A + C*sec(c + d*x)^2), x, 4, ((6*A + 5*C)*arctanh(sin(c + d*x)))/(16*d) + ((6*A + 5*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((6*A + 5*C)*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (C*sec(c + d*x)^5*tan(c + d*x))/(6*d)],
[sec(c + d*x)^4*(A + C*sec(c + d*x)^2), x, 3, ((5*A + 4*C)*tan(c + d*x))/(5*d) + (C*sec(c + d*x)^4*tan(c + d*x))/(5*d) + ((5*A + 4*C)*tan(c + d*x)^3)/(15*d)],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2), x, 3, ((4*A + 3*C)*arctanh(sin(c + d*x)))/(8*d) + ((4*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (C*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2), x, 2, ((3*A + 2*C)*tan(c + d*x))/(3*d) + (C*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2), x, 2, ((2*A + C)*arctanh(sin(c + d*x)))/(2*d) + (C*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2), x, 2, A*x + (C*tan(c + d*x))/d],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2), x, 2, (C*arctanh(sin(c + d*x)))/d + (A*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2), x, 2, (1/2)*(A + 2*C)*x + (A*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2), x, 2, ((2*A + 3*C)*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2), x, 2, (1/8)*(3*A + 4*C)*x + ((3*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2), x, 3, ((4*A + 5*C)*sin(c + d*x))/(5*d) + (A*cos(c + d*x)^4*sin(c + d*x))/(5*d) - ((4*A + 5*C)*sin(c + d*x)^3)/(15*d)],
[cos(c + d*x)^6*(A + C*sec(c + d*x)^2), x, 3, (1/16)*(5*A + 6*C)*x + ((5*A + 6*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((5*A + 6*C)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^5*sin(c + d*x))/(6*d)],

[sec(c + d*x)^m*(-C*m/(m + 1) + C*sec(c + d*x)^2), x, 1, (C*sec(c + d*x)^(1 + m)*sin(c + d*x))/(d*(1 + m))],
[sec(c + d*x)^m*(A - A*(m + 1)/m*sec(c + d*x)^2), x, 1, -((A*sec(c + d*x)^(1 + m)*sin(c + d*x))/(d*m))],


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 6, (a*(4*A + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(5*A + 4*C)*tan(c + d*x))/(15*d) + (a*(4*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(5*A + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a*C*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (C*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 5, (a*(4*A + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (a*(3*A + 2*C)*tan(c + d*x))/(3*d) + (a*(4*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*C*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(4*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (a*(2*A + C)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*A + 2*C)*tan(c + d*x))/(3*d) + (a*C*sec(c + d*x)*tan(c + d*x))/(6*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, a*A*x + (a*(2*A + C)*arctanh(sin(c + d*x)))/(2*d) + (a*C*tan(c + d*x))/(2*d) + (C*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, a*A*x + (a*C*arctanh(sin(c + d*x)))/d + (a*(A - C)*sin(c + d*x))/d + (C*(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (1/2)*a*(A + 2*C)*x + (a*C*arctanh(sin(c + d*x)))/d + (a*A*sin(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (1/2)*a*(A + 2*C)*x + (a*(2*A + 3*C)*sin(c + d*x))/(3*d) + (a*A*cos(c + d*x)*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 5, (1/8)*a*(3*A + 4*C)*x + (a*(2*A + 3*C)*sin(c + d*x))/(3*d) + (a*(3*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*A*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 6, (1/8)*a*(3*A + 4*C)*x + (2*a*(4*A + 5*C)*sin(c + d*x))/(15*d) + (a*(3*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(4*A + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a*A*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],


[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 6, (a^2*(4*A + 3*C)*arctanh(sin(c + d*x)))/(4*d) + (a^2*(25*A + 18*C)*tan(c + d*x))/(15*d) + (a^2*(4*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + (a^2*(10*A + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(30*d) + (a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(10*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (a^2*(12*A + 7*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(12*A + 7*C)*tan(c + d*x))/(6*d) + (a^2*(12*A + 7*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, a^2*A*x + (a^2*(2*A + C)*arctanh(sin(c + d*x)))/d + (a^2*(A + C)*tan(c + d*x))/d + (a*C*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, 2*a^2*A*x + (a^2*(2*A + 3*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*A - 3*C)*sin(c + d*x))/(2*d) + (a*C*(a + a*sec(c + d*x))*sin(c + d*x))/d + (C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (1/2)*a^2*(3*A + 2*C)*x + (2*a^2*C*arctanh(sin(c + d*x)))/d + (a^2*(3*A - 2*C)*sin(c + d*x))/(2*d) - (a*(A - 2*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (a^2*A + 2*a^2*C)*x + (a^2*C*arctanh(sin(c + d*x)))/d + (a^2*(A + C)*sin(c + d*x))/d + (a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (1/8)*a^2*(7*A + 12*C)*x + (a^2*(7*A + 12*C)*sin(c + d*x))/(6*d) + (a^2*(7*A + 12*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^2*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 6, (1/4)*a^2*(3*A + 4*C)*x + (a^2*(18*A + 25*C)*sin(c + d*x))/(15*d) + (a^2*(3*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^2*(9*A + 10*C)*cos(c + d*x)^2*sin(c + d*x))/(30*d) + (a*A*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(10*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 7, (1/16)*a^2*(11*A + 14*C)*x + (4*a^2*(4*A + 5*C)*sin(c + d*x))/(15*d) + (a^2*(11*A + 14*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (2*a^2*(4*A + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(9*A + 10*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(6*d)],


[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 7, (a^3*(30*A + 23*C)*arctanh(sin(c + d*x)))/(16*d) + (a^3*(45*A + 34*C)*tan(c + d*x))/(15*d) + (a^3*(30*A + 23*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(90*A + 73*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^2*(30*A + 31*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(120*d) + (a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(10*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^3*tan(c + d*x))/(6*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (a^3*(20*A + 13*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(20*A + 13*C)*tan(c + d*x))/(6*d) + (a^3*(20*A + 13*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(20*A + 13*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (3*C*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, a^3*A*x + (a^3*(28*A + 15*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(4*A + 3*C)*tan(c + d*x))/(8*d) + (a^2*(4*A + 5*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(8*d) + (a*C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d) + (C*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, 3*a^3*A*x + (a^3*(6*A + 5*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*C*sin(c + d*x))/(2*d) + (a^2*(3*A + 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (a*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (1/2)*a^3*(7*A + 2*C)*x + (a^3*(2*A + 7*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(A - C)*sin(c + d*x))/(2*d) - (a^2*(A - 4*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) - (a*(A - C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (1/2)*a^3*(5*A + 6*C)*x + (3*a^3*C*arctanh(sin(c + d*x)))/d + (5*a^3*A*sin(c + d*x))/(2*d) - (a^2*(5*A - 6*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) + (a*A*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (a^3*(15*A + 28*C)*x)/8 + (a^3*C*arctanh(sin(c + d*x)))/d + (5*a^3*(3*A + 4*C)*sin(c + d*x))/(8*d) + (a^2*(5*A + 4*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(8*d) + (a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (1/8)*a^3*(13*A + 20*C)*x + (a^3*(13*A + 20*C)*sin(c + d*x))/(6*d) + (a^3*(13*A + 20*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(13*A + 20*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (3*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 7, (1/16)*a^3*(23*A + 30*C)*x + (a^3*(34*A + 45*C)*sin(c + d*x))/(15*d) + (a^3*(23*A + 30*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(73*A + 90*C)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + (a^2*(31*A + 30*C)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(120*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(10*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d)],


[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 8, (a^4*(14*A + 11*C)*arctanh(sin(c + d*x)))/(4*d) + (a^4*(581*A + 454*C)*tan(c + d*x))/(105*d) + (a^4*(14*A + 11*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + (a^4*(308*A + 247*C)*sec(c + d*x)^2*tan(c + d*x))/(210*d) + (a^3*(126*A + 109*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(210*d) + (a^2*(7*A + 8*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(35*d) + (2*a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^3*tan(c + d*x))/(21*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^4*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, (7*a^4*(10*A + 7*C)*arctanh(sin(c + d*x)))/(16*d) + (7*a^4*(10*A + 7*C)*tan(c + d*x))/(12*d) + (7*a^4*(10*A + 7*C)*sec(c + d*x)*tan(c + d*x))/(48*d) + (7*a^2*(10*A + 7*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(120*d) + (a*(10*A + 7*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(40*d) + (2*C*(a + a*sec(c + d*x))^4*tan(c + d*x))/(15*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^4*tan(c + d*x))/(6*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, a^4*A*x + (a^4*(12*A + 7*C)*arctanh(sin(c + d*x)))/(2*d) + (a^4*(10*A + 7*C)*tan(c + d*x))/(2*d) + (a^3*(8*A + 7*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a^2*(5*A + 7*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*C*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d) + (C*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, 4*a^4*A*x + (a^4*(52*A + 35*C)*arctanh(sin(c + d*x)))/(8*d) - (5*a^4*(4*A + 7*C)*sin(c + d*x))/(8*d) + (a^3*(36*A + 35*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(12*d) + (a^2*(4*A + 7*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(8*d) + (a*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (C*(a + a*sec(c + d*x))^4*sin(c + d*x))/(4*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, (1/2)*a^4*(13*A + 2*C)*x + (2*a^4*(2*A + 3*C)*arctanh(sin(c + d*x)))/d + (5*a^4*(A - 2*C)*sin(c + d*x))/(2*d) + (a^3*(3*A + 22*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) - (a^2*(A - 2*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (a*(3*A - 2*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, 2*a^4*(3*A + 2*C)*x + (a^4*(2*A + 13*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^4*(2*A - C)*sin(c + d*x))/(2*d) - (a^3*(4*A - 9*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) - (a^2*(2*A - C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (2*a*A*cos(c + d*x)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^4*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, (1/8)*a^4*(35*A + 52*C)*x + (4*a^4*C*arctanh(sin(c + d*x)))/d + (5*a^4*(7*A + 4*C)*sin(c + d*x))/(8*d) - (a^3*(35*A - 12*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(7*A + 4*C)*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(8*d) + (a*A*cos(c + d*x)^2*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^4*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, (1/2)*a^4*(7*A + 12*C)*x + (a^4*C*arctanh(sin(c + d*x)))/d + (a^4*(7*A + 10*C)*sin(c + d*x))/(2*d) + (a^3*(7*A + 8*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a^2*(7*A + 5*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d) + (A*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 7, (7/16)*a^4*(7*A + 10*C)*x + (7*a^4*(7*A + 10*C)*sin(c + d*x))/(12*d) + (7*a^4*(7*A + 10*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (7*a^2*(7*A + 10*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(7*A + 10*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) + (2*A*(a + a*cos(c + d*x))^4*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^4*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^4, x, 8, (1/4)*a^4*(11*A + 14*C)*x + (a^4*(454*A + 581*C)*sin(c + d*x))/(105*d) + (a^4*(11*A + 14*C)*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^4*(247*A + 308*C)*cos(c + d*x)^2*sin(c + d*x))/(210*d) + (a^3*(109*A + 126*C)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(210*d) + (a^2*(8*A + 7*C)*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*a*A*cos(c + d*x)^5*(a + a*sec(c + d*x))^3*sin(c + d*x))/(21*d) + (A*cos(c + d*x)^6*(a + a*sec(c + d*x))^4*sin(c + d*x))/(7*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, (3*(4*A + 5*C)*arctanh(sin(c + d*x)))/(8*a*d) - (2*(3*A + 4*C)*tan(c + d*x))/(3*a*d) + (3*(4*A + 5*C)*sec(c + d*x)*tan(c + d*x))/(8*a*d) - ((3*A + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) + ((4*A + 5*C)*sec(c + d*x)^3*tan(c + d*x))/(4*a*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, -(((2*A + 3*C)*arctanh(sin(c + d*x)))/(2*a*d)) + (2*(3*A + 4*C)*tan(c + d*x))/(3*a*d) - ((2*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) + ((3*A + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, ((2*A + 3*C)*arctanh(sin(c + d*x)))/(2*a*d) - ((A + 2*C)*tan(c + d*x))/(a*d) + ((2*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, -((C*arctanh(sin(c + d*x)))/(a*d)) + ((A + 2*C)*tan(c + d*x))/(a*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, (A*x)/a + (C*arctanh(sin(c + d*x)))/(a*d) - ((A + C)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, -((A*x)/a) + ((2*A + C)*sin(c + d*x))/(a*d) - ((A + C)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, ((3*A + 2*C)*x)/(2*a) - ((2*A + C)*sin(c + d*x))/(a*d) + ((3*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, -(((3*A + 2*C)*x)/(2*a)) + (2*(4*A + 3*C)*sin(c + d*x))/(3*a*d) - ((3*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) + ((4*A + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, (3*(5*A + 4*C)*x)/(8*a) - (2*(4*A + 3*C)*sin(c + d*x))/(3*a*d) + (3*(5*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*a*d) - ((4*A + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) + ((5*A + 4*C)*cos(c + d*x)^3*sin(c + d*x))/(4*a*d) - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, -(((2*A + 5*C)*arctanh(sin(c + d*x)))/(a^2*d)) + (2*(5*A + 12*C)*tan(c + d*x))/(3*a^2*d) - ((2*A + 5*C)*sec(c + d*x)*tan(c + d*x))/(a^2*d) + ((5*A + 12*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*(2*A + 5*C)*sec(c + d*x)^3*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, ((2*A + 7*C)*arctanh(sin(c + d*x)))/(2*a^2*d) - (4*(A + 4*C)*tan(c + d*x))/(3*a^2*d) + ((2*A + 7*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*(A + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, -((2*C*arctanh(sin(c + d*x)))/(a^2*d)) + ((A + 10*C)*tan(c + d*x))/(3*a^2*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*C*sec(c + d*x)*tan(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 3, (C*arctanh(sin(c + d*x)))/(a^2*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + (2*(A - 2*C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 3, (A*x)/a^2 - ((A + C)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*(2*A - C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, -((2*A*x)/a^2) + ((10*A + C)*sin(c + d*x))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*A*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, ((7*A + 2*C)*x)/(2*a^2) - (4*(4*A + C)*sin(c + d*x))/(3*a^2*d) + ((7*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*(4*A + C)*cos(c + d*x)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, -(((5*A + 2*C)*x)/a^2) + (2*(12*A + 5*C)*sin(c + d*x))/(3*a^2*d) - ((5*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(a^2*d) + ((12*A + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - (2*(5*A + 2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, ((2*A + 13*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (2*(11*A + 76*C)*tan(c + d*x))/(15*a^3*d) + ((2*A + 13*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((A + 11*C)*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((11*A + 76*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, -((3*C*arctanh(sin(c + d*x)))/(a^3*d)) + (2*(A + 36*C)*tan(c + d*x))/(15*a^3*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((A - 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (3*C*sec(c + d*x)*tan(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, (C*arctanh(sin(c + d*x)))/(a^3*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((3*A - 7*C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A - 22*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 3, -(((A + C)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3)) + ((A - C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) + ((2*A + 7*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, (A*x)/a^3 - ((A + C)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((7*A - 3*C)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((22*A - 3*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, -((3*A*x)/a^3) + (2*(36*A + C)*sin(c + d*x))/(15*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((9*A - C)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (3*A*sin(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, ((13*A + 2*C)*x)/(2*a^3) - (2*(76*A + 11*C)*sin(c + d*x))/(15*a^3*d) + ((13*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((11*A + C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((76*A + 11*C)*cos(c + d*x)*sin(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 7, -(((23*A + 6*C)*x)/(2*a^3)) + (8*(34*A + 9*C)*sin(c + d*x))/(15*a^3*d) - ((23*A + 6*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (4*(34*A + 9*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((13*A + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((23*A + 6*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^5*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 7, ((2*A + 21*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - (32*(5*A + 54*C)*tan(c + d*x))/(105*a^4*d) + ((2*A + 21*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - ((A + C)*sec(c + d*x)^5*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*C*sec(c + d*x)^4*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - ((10*A + 129*C)*sec(c + d*x)^3*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (16*(5*A + 54*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 6, -((4*C*arctanh(sin(c + d*x)))/(a^4*d)) + (2*(3*A + 332*C)*tan(c + d*x))/(105*a^4*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + (2*(A - 6*C)*sec(c + d*x)^3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((3*A - 88*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*C*sec(c + d*x)*tan(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 5, (C*arctanh(sin(c + d*x)))/(a^4*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + (2*(2*A - 5*C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((8*A - 55*C)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (8*(A - 20*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 4, -(((A + C)*sec(c + d*x)^2*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) - (2*(3*A - 4*C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((13*A + 6*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (2*(13*A + 6*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 4, -(((A + C)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) + (2*(4*A - 3*C)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((6*A + 13*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + ((6*A + 13*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 5, (A*x)/a^4 - ((A + C)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*(5*A - 2*C)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((55*A - 8*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (8*(20*A - C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 6, -((4*A*x)/a^4) + (2*(332*A + 3*C)*sin(c + d*x))/(105*a^4*d) - ((A + C)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*(6*A - C)*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((88*A - 3*C)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*A*sin(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 7, ((21*A + 2*C)*x)/(2*a^4) - (32*(54*A + 5*C)*sin(c + d*x))/(105*a^4*d) + ((21*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*A*cos(c + d*x)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - ((129*A + 10*C)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (16*(54*A + 5*C)*cos(c + d*x)*sin(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 8, -((2*(11*A + 2*C)*x)/a^4) + (8*(454*A + 83*C)*sin(c + d*x))/(105*a^4*d) - (2*(11*A + 2*C)*cos(c + d*x)*sin(c + d*x))/(a^4*d) + (4*(454*A + 83*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^4*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - (2*(8*A + C)*cos(c + d*x)^2*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((178*A + 31*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*(11*A + 2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^3*d*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 6, (16*a*(99*A + 80*C)*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (4*a*(99*A + 80*C)*sec(c + d*x)^2*tan(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a*(99*A + 80*C)*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a*C*sec(c + d*x)^4*tan(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) + (16*(99*A + 80*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3465*d) + (2*C*sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(11*d)],
[sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (8*a*(21*A + 16*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(21*A + 16*C)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*C*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (8*(21*A + 16*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*C*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (2*a*(35*A + 24*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*C*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*(35*A + 24*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*C*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 3, (2*a*(15*A + 7*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*sec(c + d*x)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (2*sqrt(a)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a*C*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 3, (sqrt(a)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*(A - 2*C)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 3, (sqrt(a)*(3*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a*A*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (sqrt(a)*(5*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a*(5*A + 8*C)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (sqrt(a)*(35*A + 48*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a*(35*A + 48*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a*(35*A + 48*C)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 6, (8*a^2*(143*A + 112*C)*tan(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(143*A + 112*C)*sec(c + d*x)^2*tan(c + d*x))/(385*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(33*A + 28*C)*sec(c + d*x)^3*tan(c + d*x))/(231*d*sqrt(a + a*sec(c + d*x))) + (8*a*(143*A + 112*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(1155*d) + (2*a*C*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(33*d) + (2*C*sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (2*a^2*(189*A + 136*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(63*A + 52*C)*sec(c + d*x)^2*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(189*A + 136*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a*C*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 4, (8*a^2*(35*A + 19*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(35*A + 19*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (6*C*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*C*sec(c + d*x)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (2*a^(3/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*(5*A + 4*C)*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*C*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*C*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 4, (3*a^(3/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*(3*A - 8*C)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d + (2*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 4, (a^(3/2)*(7*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^2*(5*A - 8*C)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) - (a*(A - 4*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 4, (a^(3/2)*(11*A + 24*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^2*(19*A + 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (a^(3/2)*(75*A + 112*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^2*(75*A + 112*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(13*A + 16*C)*cos(c + d*x)*sin(c + d*x))/(32*d*sqrt(a + a*sec(c + d*x))) + (a*A*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 6, (a^(3/2)*(133*A + 176*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(128*d) + (a^2*(133*A + 176*C)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^2*(133*A + 176*C)*cos(c + d*x)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(67*A + 80*C)*cos(c + d*x)^2*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (3*a*A*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 7, (8*a^3*(10439*A + 8368*C)*tan(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(10439*A + 8368*C)*sec(c + d*x)^2*tan(c + d*x))/(15015*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2717*A + 2224*C)*sec(c + d*x)^3*tan(c + d*x))/(9009*d*sqrt(a + a*sec(c + d*x))) + (8*a^2*(10439*A + 8368*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(45045*d) + (2*a^2*(143*A + 136*C)*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(1287*d) + (10*a*C*sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(143*d) + (2*C*sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(13*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (2*a^3*(759*A + 568*C)*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(297*A + 232*C)*sec(c + d*x)^2*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(759*A + 568*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(693*d) + (2*a^2*(33*A + 32*C)*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(231*d) + (10*a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (64*a^3*(21*A + 13*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(21*A + 13*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a*(21*A + 13*C)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(105*d) + (10*C*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(63*d) + (2*C*sec(c + d*x)*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (2*a^(5/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^3*(49*A + 32*C)*tan(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(7*A + 8*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*C*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*C*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (5*a^(5/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d - (a^3*(15*A + 64*C)*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(5*A + 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(19*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^3*(27*A - 56*C)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) - (a^2*(A - 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) - (a*(3*A - 4*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (5*a^(5/2)*(5*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^3*(49*A - 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) - (a^2*(3*A - 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (5*a*A*cos(c + d*x)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(163*A + 304*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^3*(299*A + 432*C)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(17*A + 16*C)*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d) + (5*a*A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (a^(5/2)*(283*A + 400*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(128*d) + (a^3*(283*A + 400*C)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(787*A + 1040*C)*cos(c + d*x)*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(79*A + 80*C)*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d) + (a*A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 7, (a^(5/2)*(1015*A + 1304*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(512*d) + (a^3*(1015*A + 1304*C)*sin(c + d*x))/(512*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1015*A + 1304*C)*cos(c + d*x)*sin(c + d*x))/(768*d*sqrt(a + a*sec(c + d*x))) + (a^3*(109*A + 136*C)*cos(c + d*x)^2*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(23*A + 24*C)*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(96*d) + (a*A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 6, -((sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(273*A + 257*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) - (2*(21*A + 29*C)*sec(c + d*x)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*(21*A + 19*C)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*C*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^4*tan(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*(35*A + 43*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*(35*A + 31*C)*sec(c + d*x)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*C*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 4, -((sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(15*A + 13*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) - (2*C*sec(c + d*x)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*C*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*C*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, -((A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d)) + (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*C*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + ((A + 2*C)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, ((7*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (A*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 6, -((9*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*sqrt(a)*d) + (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + ((7*A + 8*C)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) - (A*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 7, ((107*A + 112*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - ((21*A + 16*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + ((43*A + 48*C)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) - (A*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 6, ((11*A + 19*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((665*A + 1201*C)*tan(c + d*x))/(210*a*d*sqrt(a + a*sec(c + d*x))) + ((245*A + 397*C)*sec(c + d*x)*tan(c + d*x))/(210*a*d*sqrt(a + a*sec(c + d*x))) - ((35*A + 67*C)*sec(c + d*x)^2*tan(c + d*x))/(70*a*d*sqrt(a + a*sec(c + d*x))) + ((7*A + 11*C)*sec(c + d*x)^3*tan(c + d*x))/(14*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 5, -((7*A + 15*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((25*A + 49*C)*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) - ((5*A + 13*C)*sec(c + d*x)*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) + ((5*A + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 4, ((3*A + 11*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((3*A + 19*C)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x))) + ((3*A + 7*C)*sec(c + d*x)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 3, ((A - 7*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((A + 5*C)*tan(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - ((5*A - 3*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 5, (-3*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) + ((9*A + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((3*A + C)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 6, ((19*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(3/2)*d) - ((13*A + 5*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((7*A + 2*C)*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + ((2*A + C)*cos(c + d*x)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 7, -((47*A + 24*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*a^(3/2)*d) + ((17*A + 9*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + (3*(7*A + 4*C)*sin(c + d*x))/(8*a*d*sqrt(a + a*sec(c + d*x))) - ((13*A + 6*C)*cos(c + d*x)*sin(c + d*x))/(12*a*d*sqrt(a + a*sec(c + d*x))) + ((5*A + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 6, -((75*A + 283*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sec(c + d*x)^4*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((5*A + 21*C)*sec(c + d*x)^3*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((735*A + 2671*C)*tan(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) - ((195*A + 787*C)*sec(c + d*x)*tan(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) + ((45*A + 157*C)*sec(c + d*x)^2*tan(c + d*x))/(80*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 5, ((19*A + 163*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sec(c + d*x)^3*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((A + 17*C)*sec(c + d*x)^2*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((27*A + 299*C)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x))) + (5*(3*A + 19*C)*sec(c + d*x)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 4, (5*(A - 15*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((3*A - 13*C)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((A + 49*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 3, ((3*A + 19*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((7*A - 9*C)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 5, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - ((43*A - 5*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((11*A - 5*C)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 6, (-5*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) + ((115*A + 3*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((15*A - C)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((35*A + 3*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 7, ((39*A + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(5/2)*d) - ((219*A + 43*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*cos(c + d*x)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((19*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((63*A + 11*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + ((31*A + 7*C)*cos(c + d*x)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(A + C*sec(c + d*x)^2)*sec(c + d*x)^(5/2), x, 4, (2*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*C*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(A + C*sec(c + d*x)^2)*sec(c + d*x)^(3/2), x, 4, -((2*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*C*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(A + C*sec(c + d*x)^2)*sec(c + d*x)^(1/2), x, 3, (2*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(A + C*sec(c + d*x)^2)/sec(c + d*x)^(1/2), x, 3, (2*(A - C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*C*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(A + C*sec(c + d*x)^2)/sec(c + d*x)^(3/2), x, 3, (2*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/sec(c + d*x)^(5/2), x, 3, (2*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(A + C*sec(c + d*x)^2)/sec(c + d*x)^(7/2), x, 4, (2*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*(5*A + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/sec(c + d*x)^(9/2), x, 4, (2*(7*A + 9*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*A*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (2*(7*A + 9*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2))],

[(3 + 3*sec(c + d*x)^2)/sec(c + d*x)^(1/2), x, 1, (6*sqrt(sec(c + d*x))*sin(c + d*x))/d],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 8, (-2*a*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(5*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(7*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*a*C*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 7, (-2*a*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(5*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 6, (2*a*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*C*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 6, (2*a*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a*(A - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 6, (2*a*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*A*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 7, (2*a*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a*A*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a*(5*A + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (2*a*(7*A + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a*A*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a*(7*A + 9*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*a*(5*A + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 9, (-16*a^2*(3*A + 2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (16*a^2*(3*A + 2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^2*(7*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a^2*(21*A + 19*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (8*a*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 8, (-4*a^2*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*(7*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(5*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(35*A + 33*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (8*a*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 7, (-16*a^2*C*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(15*A + 17*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 7, (4*a^2*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*(A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(A - 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) - (2*a*(A - C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 7, (16*a^2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(7*A - 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 7, (4*a^2*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*(3*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(33*A + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (8*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (16*a^2*(2*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(19*A + 21*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (4*a^2*(5*A + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (8*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 9, (4*a^2*(7*A + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (8*a^2*(25*A + 33*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*a^2*(89*A + 99*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (4*a^2*(7*A + 9*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (8*a^2*(25*A + 33*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (8*a*A*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 10, (-4*a^3*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(143*A + 105*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(7*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^3*(143*A + 105*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(231*d) + (8*a^3*(44*A + 35*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(385*d) + (2*a^2*(33*A + 35*C)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(231*d) + (4*a*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(33*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 9, (-4*a^3*(27*A + 17*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(21*A + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(27*A + 17*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a^3*(21*A + 16*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(63*A + 73*C)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (4*a*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d)],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 8, (-4*a^3*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(35*A + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a^3*(70*A + 53*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(5*A + 7*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (12*a*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 8, (4*a^3*(5*A - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*(5*A + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(5*A - 9*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a*(5*A - 3*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 8, (4*a^3*(9*A - 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (8*a^3*(3*A - 10*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(9*A - 5*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 8, (4*a^3*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(13*A + 35*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) - (4*a^3*(41*A - 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (12*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (4*a^3*(17*A + 27*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(11*A + 21*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a^3*(16*A + 21*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(73*A + 63*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (4*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 9, (4*a^3*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(105*A + 143*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (8*a^3*(35*A + 44*C)*sin(c + d*x))/(385*d*sec(c + d*x)^(3/2)) + (4*a^3*(105*A + 143*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(35*A + 33*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)) + (4*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(33*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(13/2), x, 10, (4*a^3*(175*A + 221*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (4*a^3*(95*A + 121*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (40*a^3*(118*A + 143*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (4*a^3*(175*A + 221*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (4*a^3*(95*A + 121*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(145*A + 143*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(7/2)) + (12*a*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(9/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[(sec(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 8, (-3*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a*d) - ((3*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + (3*(5*A + 7*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d) - ((3*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) + ((5*A + 7*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) - ((A + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 7, ((A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) + ((3*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) - ((A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + ((3*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 6, -(((A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d)) + ((A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) + ((A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))), x, 5, ((3*A + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, -(((3*A + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d)) + ((5*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((5*A + 3*C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (3*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a*d) - ((5*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((7*A + 5*C)*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - ((5*A + 3*C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 8, ((A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*(A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + 7*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*(A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - ((A + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((A + 7*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 7, (-4*C*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((A - 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*C*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A - 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 6, -(((A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*(A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2), x, 6, (4*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) - ((5*A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((5*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 7, -(((7*A + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*(5*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (2*(5*A + C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - ((7*A + C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 8, (4*(14*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*(14*A + 5*C)*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*(3*A + C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((3*A + C)*sin(c + d*x))/(a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(7/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 9, ((9*A + 119*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - ((9*A + 119*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((A + 11*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a^3*d) - ((A + C)*sec(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*C*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - ((9*A + 119*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 8, ((A - 49*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A - 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((A + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*(A - 4*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A - 13*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((A - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*(2*A - 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A - 9*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((9*A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*(3*A - 2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3), x, 7, ((49*A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*(4*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((13*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 8, -((119*A + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((11*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + ((11*A + C)*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - (2*A*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - ((119*A + 9*C)*sin(c + d*x))/(30*a^2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 9, (7*(33*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((63*A + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (7*(33*A + 7*C)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) - ((63*A + 13*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - (2*(6*A + C)*sin(c + d*x))/(15*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((63*A + 13*C)*sin(c + d*x))/(10*a^2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 6, (a*(48*A + 35*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 35*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*C*sec(c + d*x)^(7/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (a*(8*A + 5*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(8*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*C*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (a*(8*A + 3*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 4, (sqrt(2)*a*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a*(2*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (C*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 4, (2*sqrt(2)*a*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 3, (2*a*A*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(8*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 4, (2*a*A*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(24*A + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a*(24*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (2*a*A*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(16*A + 21*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a*(16*A + 21*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*(16*A + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 7, (a^2*(176*A + 133*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 133*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 133*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 67*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (3*a*C*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (C*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 6, (a^2*(112*A + 75*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(112*A + 75*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 13*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(32*d*sqrt(a + a*sec(c + d*x))) + (a*C*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(8*d) + (C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (a^2*(24*A + 11*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 19*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*C*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 5, (a^2*(8*A + 7*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (3*a*C*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 5, (3*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) - (a*(2*A - 3*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 5, (2*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 4, (8*a^2*(19*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(19*A + 35*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (6*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (2*a^2*(52*A + 63*C)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 189*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^2*(136*A + 189*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 6, (2*a^2*(28*A + 33*C)*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(112*A + 143*C)*sin(c + d*x))/(385*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^2*(112*A + 143*C)*sin(c + d*x))/(1155*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(112*A + 143*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(33*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 8, (a^3*(1304*A + 1015*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(256*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1015*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(512*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1015*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(768*d*sqrt(a + a*sec(c + d*x))) + (a^3*(136*A + 109*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 23*C)*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(96*d) + (a*C*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (C*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 7, (a^3*(400*A + 283*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(400*A + 283*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1040*A + 787*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 79*C)*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d) + (a*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (a^3*(304*A + 163*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(432*A + 299*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 17*C)*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d) + (5*a*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 6, (5*a^3*(8*A + 5*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(24*A - 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 31*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (5*a*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 6, (a^3*(8*A + 19*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(56*A - 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) - (a^2*(8*A - 21*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(12*d) - (a*(4*A - 3*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 6, (5*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^3*(64*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) - (a^2*(16*A - 15*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 6, (2*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(32*A + 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(8*A + 7*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (64*a^3*(13*A + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(13*A + 21*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*a*(13*A + 21*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (10*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 6, (2*a^3*(232*A + 297*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(568*A + 759*C)*sin(c + d*x))/(693*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^3*(568*A + 759*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(32*A + 33*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)) + (10*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sec(c + d*x)^(13/2), x, 7, (2*a^3*(2224*A + 2717*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(8368*A + 10439*C)*sin(c + d*x))/(15015*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^3*(8368*A + 10439*C)*sin(c + d*x))/(45045*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^3*(8368*A + 10439*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 143*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(7/2)) + (10*a*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(143*d*sec(c + d*x)^(9/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(13*d*sec(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[(sec(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 8, (2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) - ((8*A + 9*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + ((8*A + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) - (C*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 7, -((2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))) + ((8*A + 7*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) - (C*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) - (sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 6, -((2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 4, (2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 5, -((2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x)))) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (2*A*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(13*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - (2*A*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*(31*A + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 7, ((A + 9*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - (3*sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((A + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 6, ((3*A - 5*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 4, -(((7*A - C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((5*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 5, ((11*A + 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((7*A + 3*C)*sin(c + d*x))/(6*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((19*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, -(((15*A + 7*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) - ((A + C)*sin(c + d*x))/(2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((9*A + 5*C)*sin(c + d*x))/(10*a*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((13*A + 5*C)*sin(c + d*x))/(10*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((49*A + 25*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 8, ((3*A + 115*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - (5*sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((A - 15*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((3*A + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 7, ((5*A - 43*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((5*A - 11*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[(sqrt(sec(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 4, ((19*A + 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((9*A - 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 5, -((5*(15*A - C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((49*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 6, ((163*A + 19*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((17*A + C)*sin(c + d*x))/(16*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + (5*(19*A + 3*C)*sin(c + d*x))/(48*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((299*A + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, -(((283*A + 75*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))) - ((A + C)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - ((21*A + 5*C)*sin(c + d*x))/(16*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((157*A + 45*C)*sin(c + d*x))/(80*a^2*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((787*A + 195*C)*sin(c + d*x))/(240*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((2671*A + 735*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/3)
#


[(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(2/3), x, 1, Int((a + a*sec(c + d*x))^(2/3)*(A + C*sec(c + d*x)^2), x)],
[(A + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(1/3), x, 1, Int((a + a*sec(c + d*x))^(1/3)*(A + C*sec(c + d*x)^2), x)],
[(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(1/3), x, 0, Int((A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(1/3), x)],
[(A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(2/3), x, 0, Int((A + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(A + C*sec(c + d*x)^2)*cos(c + d*x)^(9/2), x, 3, (2*(7*A + 9*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(7*A + 9*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*A*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)],
[(A + C*sec(c + d*x)^2)*cos(c + d*x)^(7/2), x, 3, (2*(5*A + 7*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(5*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[(A + C*sec(c + d*x)^2)*cos(c + d*x)^(5/2), x, 2, (2*(3*A + 5*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[(A + C*sec(c + d*x)^2)*cos(c + d*x)^(3/2), x, 2, (2*(A + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[(A + C*sec(c + d*x)^2)*cos(c + d*x)^(1/2), x, 2, (2*(A - C)*EllipticE((1/2)*(c + d*x), 2))/d + (2*C*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(A + C*sec(c + d*x)^2)/cos(c + d*x)^(1/2), x, 2, (2*(3*A + C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*C*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[(A + C*sec(c + d*x)^2)/cos(c + d*x)^(3/2), x, 3, -((2*(5*A + 3*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*C*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(5*A + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[(A + C*sec(c + d*x)^2)/cos(c + d*x)^(5/2), x, 3, (2*(7*A + 5*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*C*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(7*A + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 7, (2*a*(7*A + 9*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*a*(5*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(5*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(7*A + 9*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 6, (2*a*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(5*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(5*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (2*a*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a*A*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (2*a*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*a*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a*(A - 3*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (2*a*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*a*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a*C*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (-2*a*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a*C*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (-2*a*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a*C*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*(7*A + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 8, (4*a^2*(7*A + 9*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a^2*(25*A + 33*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (8*a^2*(25*A + 33*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^2*(7*A + 9*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(89*A + 99*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (8*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 7, (16*a^2*(2*A + 3*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^2*(5*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(19*A + 21*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (8*a*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 6, (4*a^2*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(33*A + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (8*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 6, (16*a^2*A*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(7*A - 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 6, (4*a^2*(A - C)*EllipticE((c + d*x)/2, 2))/d + (8*a^2*(A + C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(A - 5*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) - (2*a*(A - C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2), x, 6, (-16*a^2*C*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(15*A + 17*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*a*C*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (-4*a^2*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(35*A + 33*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^2*(5*A + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (8*a*C*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(3/2)) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 8, (-16*a^2*(3*A + 2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(21*A + 19*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^2*(7*A + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (16*a^2*(3*A + 2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*a*C*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(5/2)) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 9, (4*a^3*(175*A + 221*C)*EllipticE((c + d*x)/2, 2))/(195*d) + (4*a^3*(95*A + 121*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(95*A + 121*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(175*A + 221*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(585*d) + (40*a^3*(118*A + 143*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d) + (2*a^2*(145*A + 143*C)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d) + (12*a*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 8, (4*a^3*(5*A + 7*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(105*A + 143*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (8*a^3*(35*A + 44*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(385*d) + (2*a^2*(35*A + 33*C)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(231*d) + (4*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(33*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 7, (4*a^3*(17*A + 27*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(16*A + 21*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(73*A + 63*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (4*a*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 7, (4*a^3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (4*a^3*(41*A - 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(7*A + 5*C)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (12*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 7, (4*a^3*(9*A - 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (8*a^3*(3*A - 10*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(9*A - 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (4*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 7, (4*a^3*(5*A - 9*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a^3*(5*A + 21*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(5*A - 9*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a*(5*A - 3*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2), x, 7, (-4*a^3*(5*A + 7*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(70*A + 53*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(5*A + 7*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (12*a*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 8, (-4*a^3*(27*A + 17*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(21*A + 16*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(27*A + 17*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(63*A + 73*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (4*a*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 9, (-4*a^3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (8*a^3*(44*A + 35*C)*sin(c + d*x))/(385*d*cos(c + d*x)^(5/2)) + (4*a^3*(143*A + 105*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (4*a^3*(7*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(33*A + 35*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(231*d*cos(c + d*x)^(5/2)) + (4*a*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(33*d*cos(c + d*x)^(5/2)) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d*cos(c + d*x)^(5/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 7, (-3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) + (5*(9*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*a*d) + (5*(9*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*a*d) - ((7*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) + ((9*A + 7*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*a*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 6, (3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) - ((5*A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) - ((5*A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + ((7*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 5, -(((3*A + C)*EllipticE((c + d*x)/2, 2))/(a*d)) + ((5*A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((5*A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 4, ((3*A + C)*EllipticE((c + d*x)/2, 2))/(a*d) - ((A - C)*EllipticF((c + d*x)/2, 2))/(a*d) - ((A + C)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))), x, 5, -(((A + 3*C)*EllipticE((c + d*x)/2, 2))/(a*d)) + ((A - C)*EllipticF((c + d*x)/2, 2))/(a*d) + ((A + 3*C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, ((A + 3*C)*EllipticE((c + d*x)/2, 2))/(a*d) + ((3*A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((3*A + 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - ((A + 3*C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (-3*(5*A + 7*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) - ((3*A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((5*A + 7*C)*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - ((3*A + 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) + (3*(5*A + 7*C)*sin(c + d*x))/(5*a*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(7/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 8, (-7*(11*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) + (10*(15*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*a^2*d) + (10*(15*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*a^2*d) - (7*(11*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) + (2*(15*A + 7*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*a^2*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((11*A + 5*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 7, (4*(14*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (5*(3*A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + (4*(14*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((3*A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 6, -(((7*A + C)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + (2*(5*A + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (2*(5*A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((7*A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 5, (4*A*EllipticE((c + d*x)/2, 2))/(a^2*d) - ((5*A - C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((5*A - C)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2), x, 5, -(((A - C)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + (2*(A + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) + ((A - C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 6, (-4*C*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((A - 5*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (4*C*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((A - 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 7, ((A + 7*C)*EllipticE((c + d*x)/2, 2))/(a^2*d) + (2*(A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (2*(A + 5*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - ((A + 7*C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - ((A + 7*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2), x, 8, (-4*(5*A + 14*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (4*(5*A + 14*C)*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(5/2)) - (5*(A + 3*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + (4*(5*A + 14*C)*sin(c + d*x))/(5*a^2*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2) - ((A + 3*C)*sin(c + d*x))/(a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 8, (7*(33*A + 7*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((63*A + 13*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((63*A + 13*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) + (7*(33*A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*(6*A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((63*A + 13*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((119*A + 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((11*A + C)*EllipticF((c + d*x)/2, 2))/(2*a^3*d) + ((11*A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - ((119*A + 9*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 6, ((49*A - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((13*A - C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - (2*(4*A - C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((13*A - C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3), x, 6, -((9*A - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A + C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) + (2*(3*A - 2*C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((3*A + C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 6, -((A - 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 3*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) + (2*(2*A - 3*C)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) + ((A - 9*C)*sin(c + d*x))/(10*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 7, ((A - 49*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A - 13*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A - 49*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) + (2*(A - 4*C)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((A - 13*C)*sin(c + d*x))/(6*a^2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3), x, 8, ((9*A + 119*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 11*C)*EllipticF((c + d*x)/2, 2))/(2*a^3*d) + ((A + 11*C)*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - ((9*A + 119*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) - (2*C*sin(c + d*x))/(3*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - ((9*A + 119*C)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(9/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 5, (16*a*(16*A + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a*(16*A + 21*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(16*A + 21*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (4*a*(24*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(24*A + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 3, (2*a*(8*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (2*sqrt(2)*a*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2), x, 4, (sqrt(2)*a*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(2*A - C)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 4, (a*(8*A + 3*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*C*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 5, (a*(8*A + 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*C*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(8*A + 5*C)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2))],
[(sqrt(a + a*sec(c + d*x))*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 6, (a*(48*A + 35*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*C*sin(c + d*x))/(24*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 35*C)*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 35*C)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 6, (16*a^2*(112*A + 143*C)*sin(c + d*x))/(1155*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^2*(112*A + 143*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(112*A + 143*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(385*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(28*A + 33*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(231*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(33*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (4*a^2*(136*A + 189*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 189*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(52*A + 63*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*A*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 4, (8*a^2*(19*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(19*A + 35*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d) + (6*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (2*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(4*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (3*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 3*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a*(2*A - 3*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2), x, 5, (a^2*(8*A + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 5*C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (3*a*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 5, (a^2*(24*A + 11*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 19*C)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 6, (a^2*(112*A + 75*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 13*C)*sin(c + d*x))/(32*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(112*A + 75*C)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(8*d*cos(c + d*x)^(5/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (a^2*(176*A + 133*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 67*C)*sin(c + d*x))/(240*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 133*C)*sin(c + d*x))/(192*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 133*C)*sin(c + d*x))/(128*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (3*a*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d*cos(c + d*x)^(7/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 7, (16*a^3*(8368*A + 10439*C)*sin(c + d*x))/(45045*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^3*(8368*A + 10439*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(8368*A + 10439*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15015*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2224*A + 2717*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 143*C)*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d) + (10*a*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (4*a^3*(568*A + 759*C)*sin(c + d*x))/(693*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(568*A + 759*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(232*A + 297*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(32*A + 33*C)*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(231*d) + (10*a*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 5, (64*a^3*(13*A + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(13*A + 21*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(13*A + 21*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(105*d) + (10*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (2*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(32*A + 49*C)*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(8*A + 7*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*a*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (5*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(64*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a^2*(16*A - 15*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (a^3*(8*A + 19*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(56*A - 27*C)*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a^2*(8*A - 21*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) - (a*(4*A - 3*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2), x, 6, (5*a^3*(8*A + 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(24*A - 49*C)*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 31*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))) + (5*a*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (a^3*(304*A + 163*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(432*A + 299*C)*sin(c + d*x))/(192*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 17*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d*cos(c + d*x)^(3/2)) + (5*a*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (a^3*(400*A + 283*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(1040*A + 787*C)*sin(c + d*x))/(960*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(400*A + 283*C)*sin(c + d*x))/(128*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 79*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d*cos(c + d*x)^(5/2)) + (a*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d*cos(c + d*x)^(5/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 8, (a^3*(1304*A + 1015*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(256*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(136*A + 109*C)*sin(c + d*x))/(192*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1015*C)*sin(c + d*x))/(768*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1015*C)*sin(c + d*x))/(512*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 23*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(96*d*cos(c + d*x)^(7/2)) + (a*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d*cos(c + d*x)^(7/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d*cos(c + d*x)^(7/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(31*A + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 5, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(13*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 4, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*A*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))), x, 6, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 7, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((8*A + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - (C*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 8, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((8*A + 9*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) - (C*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + ((8*A + 7*C)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],

[(sqrt(cos(c + d*x))*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(a*A - A*b - a*B + b*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*b*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 6, -((15*A + 7*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((49*A + 25*C)*sin(c + d*x))/(10*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((13*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x))) + ((9*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 5, ((11*A + 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((19*A + 3*C)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((7*A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 4, -((7*A - C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((5*A + C)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)), x, 6, ((3*A - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 7, ((A + 9*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (3*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + ((A + 3*C)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 8, -((5*A + 13*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((8*A + 19*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)) + ((A + 2*C)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - ((2*A + 7*C)*sin(c + d*x))/(4*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 7, -((283*A + 75*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((21*A + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((2671*A + 735*C)*sin(c + d*x))/(240*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((787*A + 195*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) + ((157*A + 45*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 6, ((163*A + 19*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((17*A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((299*A + 27*C)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (5*(19*A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 5, (-5*(15*A - C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 3*C)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((49*A + C)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)), x, 4, ((19*A + 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - ((9*A - 7*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, ((5*A - 43*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + ((5*A - 11*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 8, ((3*A + 115*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (5*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)) + ((A - 15*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + ((3*A + 35*C)*sin(c + d*x))/(16*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


# ::Section:: 
#Sec[c+d x]^m (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>=0


[sec(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (3*C*arctanh(sin(c + d*x)))/(8*d) + (2*B*tan(c + d*x))/(3*d) + (3*C*sec(c + d*x)*tan(c + d*x))/(8*d) + (B*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (C*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (B*arctanh(sin(c + d*x)))/(2*d) + (2*C*tan(c + d*x))/(3*d) + (B*sec(c + d*x)*tan(c + d*x))/(2*d) + (C*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (C*arctanh(sin(c + d*x)))/(2*d) + (B*tan(c + d*x))/d + (C*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (B*arctanh(sin(c + d*x)))/d + (C*tan(c + d*x))/d],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, B*x + (C*arctanh(sin(c + d*x)))/d],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, C*x + (B*sin(c + d*x))/d],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (B*x)/2 + (C*sin(c + d*x))/d + (B*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (C*x)/2 + (2*B*sin(c + d*x))/(3*d) + (C*cos(c + d*x)*sin(c + d*x))/(2*d) + (B*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^5*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (3*B*x)/8 + (2*C*sin(c + d*x))/(3*d) + (3*B*cos(c + d*x)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (B*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^6*(B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (3*C*x)/8 + (8*B*sin(c + d*x))/(15*d) + (3*C*cos(c + d*x)*sin(c + d*x))/(8*d) + (4*B*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (C*cos(c + d*x)^3*sin(c + d*x))/(4*d) + (B*cos(c + d*x)^4*sin(c + d*x))/(5*d)],


[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 6, (a*(4*B + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(B + C)*tan(c + d*x))/(3*d) + (a*(4*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(B + C)*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a*C*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 5, (a*(B + C)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*B + 2*C)*tan(c + d*x))/(3*d) + (a*(B + C)*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*C*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (a*(2*B + C)*arctanh(sin(c + d*x)))/(2*d) + (a*(2*B + C)*tan(c + d*x))/(2*d) + (C*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 3, a*B*x + (a*(B + C)*arctanh(sin(c + d*x)))/d + (a*C*tan(c + d*x))/d],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (a*B + a*C)*x + (a*C*arctanh(sin(c + d*x)))/d + (a*B*sin(c + d*x))/d],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 4, (1/2)*a*(B + 2*C)*x + (a*(B + C)*sin(c + d*x))/d + (a*B*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 5, (1/2)*a*(B + C)*x + (a*(2*B + 3*C)*sin(c + d*x))/(3*d) + (a*(B + C)*cos(c + d*x)*sin(c + d*x))/(2*d) + (a*B*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^5*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x)), x, 6, (1/8)*a*(3*B + 4*C)*x + (2*a*(B + C)*sin(c + d*x))/(3*d) + (a*(3*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(B + C)*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (a*B*cos(c + d*x)^3*sin(c + d*x))/(4*d)],


[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 7, (a^2*(7*B + 6*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a^2*(10*B + 9*C)*tan(c + d*x))/(15*d) + (a^2*(7*B + 6*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(10*B + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a^2*(5*B + 6*C)*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*C*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 6, (a^2*(8*B + 7*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(5*B + 4*C)*tan(c + d*x))/(3*d) + (a^2*(8*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(4*B + 5*C)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (a^2*(3*B + 2*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(3*B + 2*C)*tan(c + d*x))/(2*d) + (a*(3*B + 2*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 4, a^2*B*x + (a^2*(4*B + 3*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*B + 3*C)*tan(c + d*x))/(2*d) + (a*C*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (2*a^2*B + a^2*C)*x + (a^2*(B + 2*C)*arctanh(sin(c + d*x)))/d + (a^2*(B - C)*sin(c + d*x))/d + (a*C*(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (1/2)*a^2*(3*B + 4*C)*x + (a^2*C*arctanh(sin(c + d*x)))/d + (a^2*(3*B + 2*C)*sin(c + d*x))/(2*d) + (a*B*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 5, (1/2)*a^2*(2*B + 3*C)*x + (2*a^2*(2*B + 3*C)*sin(c + d*x))/(3*d) + (a^2*(2*B + 3*C)*cos(c + d*x)*sin(c + d*x))/(6*d) + (B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^5*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 6, (1/8)*a^2*(7*B + 8*C)*x + (a^2*(4*B + 5*C)*sin(c + d*x))/(3*d) + (a^2*(7*B + 8*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(5*B + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (a*B*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^6*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^2, x, 7, (1/8)*a^2*(6*B + 7*C)*x + (2*a^2*(9*B + 10*C)*sin(c + d*x))/(15*d) + (a^2*(6*B + 7*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(9*B + 10*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(6*B + 5*C)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (a*B*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],


[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 7, (a^3*(15*B + 13*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(45*B + 38*C)*tan(c + d*x))/(15*d) + (a^3*(15*B + 13*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*(45*B + 43*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a^2*(5*B + 7*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(20*d) + (a*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (5*a^3*(4*B + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(4*B + 3*C)*tan(c + d*x))/(8*d) + (5*a^2*(4*B + 3*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a*(4*B + 3*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (C*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 5, a^3*B*x + (a^3*(7*B + 5*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(B + C)*tan(c + d*x))/(2*d) + (a^2*(3*B + 5*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (3*a^3*B + a^3*C)*x + (a^3*(6*B + 7*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*C*sin(c + d*x))/(2*d) + (a^2*(B + 2*C)*(a + a*sec(c + d*x))*sin(c + d*x))/d + (a*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (1/2)*a^3*(7*B + 6*C)*x + (a^3*(B + 3*C)*arctanh(sin(c + d*x)))/d + (5*a^3*B*sin(c + d*x))/(2*d) - (a^2*(B - 2*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (a*B*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (1/2)*a^3*(5*B + 7*C)*x + (a^3*C*arctanh(sin(c + d*x)))/d + (5*a^3*(B + C)*sin(c + d*x))/(2*d) + (a^2*(5*B + 3*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a*B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^5*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 6, (5/8)*a^3*(3*B + 4*C)*x + (5*a^3*(3*B + 4*C)*sin(c + d*x))/(6*d) + (5*a^3*(3*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(3*B + 4*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (B*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^6*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 7, (1/8)*a^3*(13*B + 15*C)*x + (a^3*(38*B + 45*C)*sin(c + d*x))/(15*d) + (a^3*(13*B + 15*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*(43*B + 45*C)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + (a^2*(7*B + 5*C)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(20*d) + (a*B*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^7*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^3, x, 8, (1/16)*a^3*(23*B + 26*C)*x + (2*a^3*(17*B + 19*C)*sin(c + d*x))/(15*d) + (a^3*(23*B + 26*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(17*B + 19*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^3*(21*B + 22*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a^2*(4*B + 3*C)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (a*B*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(6*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, (3*(B - C)*arctanh(sin(c + d*x)))/(2*a*d) - (2*(3*B - 4*C)*tan(c + d*x))/(3*a*d) + (3*(B - C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) - ((3*B - 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) + ((B - C)*sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, -(((2*B - 3*C)*arctanh(sin(c + d*x)))/(2*a*d)) + (2*(B - C)*tan(c + d*x))/(a*d) - ((2*B - 3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) + ((B - C)*sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, ((B - C)*arctanh(sin(c + d*x)))/(a*d) - ((B - 2*C)*tan(c + d*x))/(a*d) + ((B - C)*sec(c + d*x)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, (C*arctanh(sin(c + d*x)))/(a*d) + ((B - C)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, (B*x)/a - ((B - C)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, -(((B - C)*x)/a) + ((2*B - C)*sin(c + d*x))/(a*d) - ((B - C)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, ((3*B - 2*C)*x)/(2*a) - (2*(B - C)*sin(c + d*x))/(a*d) + ((3*B - 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) - ((B - C)*cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, -((3*(B - C)*x)/(2*a)) + (2*(4*B - 3*C)*sin(c + d*x))/(3*a*d) - (3*(B - C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) + ((4*B - 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) - ((B - C)*cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, -(((4*B - 7*C)*arctanh(sin(c + d*x)))/(2*a^2*d)) + (2*(5*B - 8*C)*tan(c + d*x))/(3*a^2*d) - ((4*B - 7*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + ((B - C)*sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((5*B - 8*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, ((B - 2*C)*arctanh(sin(c + d*x)))/(a^2*d) - (2*(2*B - 5*C)*tan(c + d*x))/(3*a^2*d) + ((B - C)*sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((B - 2*C)*sec(c + d*x)*tan(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, (C*arctanh(sin(c + d*x)))/(a^2*d) + ((B - C)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((B - 4*C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 3, ((B + 2*C)*tan(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) + ((B - C)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, (B*x)/a^2 - ((B - C)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((4*B - C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, -(((2*B - C)*x)/a^2) + (2*(5*B - 2*C)*sin(c + d*x))/(3*a^2*d) - ((B - C)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((2*B - C)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, ((7*B - 4*C)*x)/(2*a^2) - (2*(8*B - 5*C)*sin(c + d*x))/(3*a^2*d) + ((7*B - 4*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - ((B - C)*cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((8*B - 5*C)*cos(c + d*x)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 7, -(((10*B - 7*C)*x)/(2*a^2)) + (8*(3*B - 2*C)*sin(c + d*x))/(3*a^2*d) - ((10*B - 7*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + (4*(3*B - 2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) - ((B - C)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((10*B - 7*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 7, -(((6*B - 13*C)*arctanh(sin(c + d*x)))/(2*a^3*d)) + (8*(9*B - 19*C)*tan(c + d*x))/(15*a^3*d) - ((6*B - 13*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) + ((B - C)*sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((6*B - 11*C)*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (4*(9*B - 19*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, ((B - 3*C)*arctanh(sin(c + d*x)))/(a^3*d) - (2*(11*B - 36*C)*tan(c + d*x))/(15*a^3*d) + ((B - C)*sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((4*B - 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((B - 3*C)*sec(c + d*x)*tan(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, (C*arctanh(sin(c + d*x)))/(a^3*d) + ((B - C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((2*B - 7*C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*(B - 11*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, -(((B - C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3)) - ((3*B + 2*C)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (2*(3*B + 2*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, ((2*B + 3*C)*tan(c + d*x))/(15*a^3*d*(1 + sec(c + d*x))^2) + ((2*B + 3*C)*tan(c + d*x))/(15*a^3*d*(1 + sec(c + d*x))) + ((B - C)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, (B*x)/a^3 - ((B - C)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((7*B - 2*C)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (2*(11*B - C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, -(((3*B - C)*x)/a^3) + (2*(36*B - 11*C)*sin(c + d*x))/(15*a^3*d) - ((B - C)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((9*B - 4*C)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((3*B - C)*sin(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 7, ((13*B - 6*C)*x)/(2*a^3) - (8*(19*B - 9*C)*sin(c + d*x))/(15*a^3*d) + ((13*B - 6*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((B - C)*cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((11*B - 6*C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (4*(19*B - 9*C)*cos(c + d*x)*sin(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


[(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(5/2), x, 5, (64*a^3*(7*B + 5*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(7*B + 5*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*a*(7*B + 5*C)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*C*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(3/2), x, 4, (8*a^2*(5*B + 3*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*(5*B + 3*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(1/2), x, 3, (2*a*(3*B + C)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],


[(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(1/2), x, 3, (sqrt(2)*(B - C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*C*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 3, ((B + 3*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((B - C)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 4, ((3*B + 5*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((B - C)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((3*B + 5*C)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(B*sec(c + d*x) + C*sec(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, -((6*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*C*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*C*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*sec(c + d*x)^(1/2), x, 6, -((2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(1/2), x, 5, -((2*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*C*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(3/2), x, 5, (2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(5/2), x, 6, (2*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(7/2), x, 7, (6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*C*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(9/2), x, 8, (6*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (10*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*B*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*C*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*B*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


# ::Subsubsection:: 
#n<0


# ::Subsection:: 
#Integrands of the form Sec[c+d x]^(m/2) (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)
#


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(9/2), x, 7, (6*C*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (10*B*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (10*B*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*B*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(7/2), x, 6, (6*B*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*C*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(5/2), x, 5, (2*C*EllipticE((1/2)*(c + d*x), 2))/d + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(3/2), x, 4, (2*B*EllipticE((1/2)*(c + d*x), 2))/d + (2*C*EllipticF((1/2)*(c + d*x), 2))/d],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(1/2), x, 4, -((2*C*EllipticE((1/2)*(c + d*x), 2))/d) + (2*B*EllipticF((1/2)*(c + d*x), 2))/d + (2*C*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(1/2), x, 5, -((2*B*EllipticE((1/2)*(c + d*x), 2))/d) + (2*C*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*C*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(3/2), x, 6, -((6*C*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*C*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*C*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


# ::Subsubsection:: 
#n<0


# ::Subsection:: 
#Integrands of the form Cos[c+d x]^(m/2) (B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


# ::Section:: 
#Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>=0


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (3*B*arctanh(sin(c + d*x)))/(8*d) + (2*(5*A + 4*C)*tan(c + d*x))/(15*d) + (3*B*sec(c + d*x)*tan(c + d*x))/(8*d) + ((5*A + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (B*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (C*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, ((4*A + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (2*B*tan(c + d*x))/(3*d) + ((4*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (B*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (C*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (B*arctanh(sin(c + d*x)))/(2*d) + ((3*A + 2*C)*tan(c + d*x))/(3*d) + (B*sec(c + d*x)*tan(c + d*x))/(2*d) + (C*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, ((2*A + C)*arctanh(sin(c + d*x)))/(2*d) + (B*tan(c + d*x))/d + (C*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, A*x + (B*arctanh(sin(c + d*x)))/d + (C*tan(c + d*x))/d],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, B*x + (C*arctanh(sin(c + d*x)))/d + (A*sin(c + d*x))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (1/2)*(A + 2*C)*x + (B*sin(c + d*x))/d + (A*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (B*x)/2 + ((2*A + 3*C)*sin(c + d*x))/(3*d) + (B*cos(c + d*x)*sin(c + d*x))/(2*d) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, ((3*A + 4*C)*x)/8 + (2*B*sin(c + d*x))/(3*d) + ((3*A + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (B*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (3*B*x)/8 + (2*(4*A + 5*C)*sin(c + d*x))/(15*d) + (3*B*cos(c + d*x)*sin(c + d*x))/(8*d) + ((4*A + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (B*cos(c + d*x)^3*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/16)*(5*A + 6*C)*x + (8*B*sin(c + d*x))/(15*d) + ((5*A + 6*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (4*B*cos(c + d*x)^2*sin(c + d*x))/(15*d) + ((5*A + 6*C)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (B*cos(c + d*x)^4*sin(c + d*x))/(5*d) + (A*cos(c + d*x)^5*sin(c + d*x))/(6*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a*(4*A + 3*B + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(5*A + 5*B + 4*C)*tan(c + d*x))/(15*d) + (a*(4*A + 3*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(5*A + 5*B + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a*(5*B + C)*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (C*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a*(4*A + 4*B + 3*C)*arctanh(sin(c + d*x)))/(8*d) + (a*(3*A + 2*B + 2*C)*tan(c + d*x))/(3*d) + (a*(4*A + 4*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(4*B + C)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(4*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a*(2*A + B + C)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*A + 3*B + 2*C)*tan(c + d*x))/(3*d) + (a*(3*B + C)*sec(c + d*x)*tan(c + d*x))/(6*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, a*A*x + (a*(2*A + 2*B + C)*arctanh(sin(c + d*x)))/(2*d) + (a*(2*B + C)*tan(c + d*x))/(2*d) + (C*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a*A + a*B)*x + (a*(B + C)*arctanh(sin(c + d*x)))/d + (a*(A - C)*sin(c + d*x))/d + (C*(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (1/2)*a*(A + 2*B + 2*C)*x + (a*C*arctanh(sin(c + d*x)))/d + (a*(A + 2*B)*sin(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (1/2)*a*(A + B + 2*C)*x + (a*(2*A + 3*B + 3*C)*sin(c + d*x))/(3*d) + (a*(A + 3*B)*cos(c + d*x)*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (1/8)*a*(3*A + 4*B + 4*C)*x + (a*(2*A + 2*B + 3*C)*sin(c + d*x))/(3*d) + (a*(3*A + 4*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(A + 4*B)*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/8)*a*(3*A + 3*B + 4*C)*x + (2*a*(4*A + 5*B + 5*C)*sin(c + d*x))/(15*d) + (a*(3*A + 3*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(4*A + 5*B + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a*(A + 5*B)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (a^2*(14*A + 12*B + 11*C)*arctanh(sin(c + d*x)))/(16*d) + (2*a^2*(10*A + 9*B + 8*C)*tan(c + d*x))/(15*d) + (a^2*(14*A + 12*B + 11*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^2*(10*A + 9*B + 8*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a^2*(10*A + 12*B + 9*C)*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (a*(3*B + C)*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (C*sec(c + d*x)^3*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^2*(8*A + 7*B + 6*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(25*A + 20*B + 18*C)*tan(c + d*x))/(15*d) + (a^2*(8*A + 7*B + 6*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(20*A + 25*B + 18*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a*(5*B + 2*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(20*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^2*(12*A + 8*B + 7*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(12*A + 8*B + 7*C)*tan(c + d*x))/(6*d) + (a^2*(12*A + 8*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + ((2*B + C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, a^2*A*x + (a^2*(4*A + 3*B + 2*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*A + 3*B + 2*C)*tan(c + d*x))/(2*d) + (a*(3*B + 2*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (C*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a^2*A + a^2*B)*x + (a^2*(2*A + 4*B + 3*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*A - 2*B - 3*C)*sin(c + d*x))/(2*d) + (a*(B + C)*(a + a*sec(c + d*x))*sin(c + d*x))/d + (C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (1/2)*a^2*(3*A + 4*B + 2*C)*x + (a^2*(B + 2*C)*arctanh(sin(c + d*x)))/d + (a^2*(3*A + 2*B - 2*C)*sin(c + d*x))/(2*d) - (a*(A - 2*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^2*(2*A + 3*B + 4*C)*x)/2 + (a^2*C*arctanh(sin(c + d*x)))/d + (a^2*(2*A + 3*B + 2*C)*sin(c + d*x))/(2*d) + (a*(2*A + 3*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (1/8)*a^2*(7*A + 8*B + 12*C)*x + (a^2*(7*A + 8*B + 12*C)*sin(c + d*x))/(6*d) + (a^2*(7*A + 8*B + 12*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((A + 2*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^2*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/8)*a^2*(6*A + 7*B + 8*C)*x + (a^2*(18*A + 20*B + 25*C)*sin(c + d*x))/(15*d) + (a^2*(6*A + 7*B + 8*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(18*A + 25*B + 20*C)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + (a*(2*A + 5*B)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(20*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/16)*a^2*(11*A + 12*B + 14*C)*x + (2*a^2*(8*A + 9*B + 10*C)*sin(c + d*x))/(15*d) + (a^2*(11*A + 12*B + 14*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*(8*A + 9*B + 10*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(9*A + 12*B + 10*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a*(A + 3*B)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(6*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (a^3*(26*A + 23*B + 21*C)*arctanh(sin(c + d*x)))/(16*d) + (2*a^3*(133*A + 119*B + 108*C)*tan(c + d*x))/(105*d) + (a^3*(26*A + 23*B + 21*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(133*A + 119*B + 108*C)*sec(c + d*x)^2*tan(c + d*x))/(105*d) + (a^3*(154*A + 147*B + 129*C)*sec(c + d*x)^3*tan(c + d*x))/(280*d) + (a^2*(3*A + 4*B + 3*C)*sec(c + d*x)^3*(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (a*(7*B + 3*C)*sec(c + d*x)^3*(a + a*sec(c + d*x))^2*tan(c + d*x))/(42*d) + (C*sec(c + d*x)^3*(a + a*sec(c + d*x))^3*tan(c + d*x))/(7*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (a^3*(30*A + 26*B + 23*C)*arctanh(sin(c + d*x)))/(16*d) + (a^3*(45*A + 38*B + 34*C)*tan(c + d*x))/(15*d) + (a^3*(30*A + 26*B + 23*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(90*A + 86*B + 73*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^2*(30*A + 42*B + 31*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(120*d) + (a*(2*B + C)*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(10*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^3*tan(c + d*x))/(6*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^3*(20*A + 15*B + 13*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(20*A + 15*B + 13*C)*tan(c + d*x))/(6*d) + (a^3*(20*A + 15*B + 13*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(20*A + 15*B + 13*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + ((5*B + 3*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, a^3*A*x + (a^3*(28*A + 20*B + 15*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(4*A + 4*B + 3*C)*tan(c + d*x))/(8*d) + (a^2*(12*A + 20*B + 15*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a*(4*B + 3*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (C*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (3*a^3*A + a^3*B)*x + (a^3*(6*A + 7*B + 5*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*(B + C)*sin(c + d*x))/(2*d) + (a^2*(3*A + 6*B + 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (a*(B + C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/2)*a^3*(7*A + 6*B + 2*C)*x + (a^3*(2*A + 6*B + 7*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(A - C)*sin(c + d*x))/(2*d) - (a^2*(A - 2*B - 4*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) - (a*(A - C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/2)*a^3*(5*A + 7*B + 6*C)*x + (a^3*(B + 3*C)*arctanh(sin(c + d*x)))/d + (5*a^3*(A + B)*sin(c + d*x))/(2*d) - (a^2*(5*A + 3*B - 6*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) + (a*(A + B)*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/8)*a^3*(15*A + 20*B + 28*C)*x + (a^3*C*arctanh(sin(c + d*x)))/d + (5*a^3*(3*A + 4*B + 4*C)*sin(c + d*x))/(8*d) + (a^2*(15*A + 20*B + 12*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a*(3*A + 4*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (1/8)*a^3*(13*A + 15*B + 20*C)*x + (a^3*(13*A + 15*B + 20*C)*sin(c + d*x))/(6*d) + (a^3*(13*A + 15*B + 20*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(13*A + 15*B + 20*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((3*A + 5*B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/16)*a^3*(23*A + 26*B + 30*C)*x + (a^3*(34*A + 38*B + 45*C)*sin(c + d*x))/(15*d) + (a^3*(23*A + 26*B + 30*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(73*A + 86*B + 90*C)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + (a^2*(31*A + 42*B + 30*C)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(120*d) + (a*(A + 2*B)*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(10*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (1/16)*a^3*(21*A + 23*B + 26*C)*x + (2*a^3*(108*A + 119*B + 133*C)*sin(c + d*x))/(105*d) + (a^3*(21*A + 23*B + 26*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(108*A + 119*B + 133*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^3*(129*A + 147*B + 154*C)*cos(c + d*x)^3*sin(c + d*x))/(280*d) + (a^2*(3*A + 4*B + 3*C)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (a*(3*A + 7*B)*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(42*d) + (A*cos(c + d*x)^6*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],


[sec(c + d*x)^2*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (a^4*(56*A + 49*B + 44*C)*arctanh(sin(c + d*x)))/(16*d) + (a^4*(581*A + 504*B + 454*C)*tan(c + d*x))/(105*d) + (a^4*(56*A + 49*B + 44*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^4*(1232*A + 1113*B + 988*C)*sec(c + d*x)^2*tan(c + d*x))/(840*d) + (a^3*(504*A + 511*B + 436*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))*tan(c + d*x))/(840*d) + (a^2*(14*A + 21*B + 16*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(70*d) + (a*(7*B + 4*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))^3*tan(c + d*x))/(42*d) + (C*sec(c + d*x)^2*(a + a*sec(c + d*x))^4*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (7*a^4*(10*A + 8*B + 7*C)*arctanh(sin(c + d*x)))/(16*d) + (7*a^4*(10*A + 8*B + 7*C)*tan(c + d*x))/(12*d) + (7*a^4*(10*A + 8*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(48*d) + (7*a^2*(10*A + 8*B + 7*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(120*d) + (a*(10*A + 8*B + 7*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(40*d) + ((3*B + 2*C)*(a + a*sec(c + d*x))^4*tan(c + d*x))/(15*d) + (C*sec(c + d*x)*(a + a*sec(c + d*x))^4*tan(c + d*x))/(6*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, a^4*A*x + (a^4*(48*A + 35*B + 28*C)*arctanh(sin(c + d*x)))/(8*d) + (a^4*(40*A + 35*B + 28*C)*tan(c + d*x))/(8*d) + (a^3*(32*A + 35*B + 28*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a^2*(20*A + 35*B + 28*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (a*(5*B + 4*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (C*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^4*A + a^4*B)*x + (a^4*(52*A + 48*B + 35*C)*arctanh(sin(c + d*x)))/(8*d) - (5*a^4*(4*A + 8*B + 7*C)*sin(c + d*x))/(8*d) + (a^3*(36*A + 44*B + 35*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(12*d) + (a^2*(4*A + 8*B + 7*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(8*d) + (a*(B + C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (C*(a + a*sec(c + d*x))^4*sin(c + d*x))/(4*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/2)*a^4*(13*A + 8*B + 2*C)*x + (a^4*(8*A + 13*B + 12*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^4*(A - B - 2*C)*sin(c + d*x))/(2*d) + (a^3*(3*A + 18*B + 22*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) - (a^2*(A - B - 2*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (a*(3*A - 2*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/2)*a^4*(12*A + 13*B + 8*C)*x + (a^4*(2*A + 8*B + 13*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^4*(2*A + B - C)*sin(c + d*x))/(2*d) - (a^3*(8*A - 3*B - 18*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(6*d) - (a^2*(2*A + B - C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*(4*A + 3*B)*cos(c + d*x)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^4*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/8)*a^4*(35*A + 48*B + 52*C)*x + (a^4*(B + 4*C)*arctanh(sin(c + d*x)))/d + (5*a^4*(7*A + 8*B + 4*C)*sin(c + d*x))/(8*d) - (a^3*(35*A + 32*B - 12*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(7*A + 8*B + 4*C)*cos(c + d*x)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(8*d) + (a*(A + B)*cos(c + d*x)^2*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^4*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (1/8)*a^4*(28*A + 35*B + 48*C)*x + (a^4*C*arctanh(sin(c + d*x)))/d + (a^4*(28*A + 35*B + 40*C)*sin(c + d*x))/(8*d) + (a^3*(28*A + 35*B + 32*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(28*A + 35*B + 20*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (a*(4*A + 5*B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (A*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (7/16)*a^4*(7*A + 8*B + 10*C)*x + (7*a^4*(7*A + 8*B + 10*C)*sin(c + d*x))/(12*d) + (7*a^4*(7*A + 8*B + 10*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (7*a^2*(7*A + 8*B + 10*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(7*A + 8*B + 10*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) + ((2*A + 3*B)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^4*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (1/16)*a^4*(44*A + 49*B + 56*C)*x + (a^4*(454*A + 504*B + 581*C)*sin(c + d*x))/(105*d) + (a^4*(44*A + 49*B + 56*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*cos(c + d*x)^2*sin(c + d*x))/(840*d) + (a^3*(436*A + 511*B + 504*C)*cos(c + d*x)^3*(a + a*sec(c + d*x))*sin(c + d*x))/(840*d) + (a^2*(16*A + 21*B + 14*C)*cos(c + d*x)^4*(a + a*sec(c + d*x))^2*sin(c + d*x))/(70*d) + (a*(4*A + 7*B)*cos(c + d*x)^5*(a + a*sec(c + d*x))^3*sin(c + d*x))/(42*d) + (A*cos(c + d*x)^6*(a + a*sec(c + d*x))^4*sin(c + d*x))/(7*d)],
[cos(c + d*x)^8*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (1/128)*a^4*(323*A + 352*B + 392*C)*x + (2*a^4*(208*A + 227*B + 252*C)*sin(c + d*x))/(105*d) + (a^4*(323*A + 352*B + 392*C)*cos(c + d*x)*sin(c + d*x))/(128*d) + (a^4*(208*A + 227*B + 252*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^4*(2007*A + 2208*B + 2408*C)*cos(c + d*x)^3*sin(c + d*x))/(2240*d) + (7*a^3*(7*A + 8*B + 8*C)*cos(c + d*x)^4*(a + a*sec(c + d*x))*sin(c + d*x))/(120*d) + (a^2*(61*A + 80*B + 56*C)*cos(c + d*x)^5*(a + a*sec(c + d*x))^2*sin(c + d*x))/(336*d) + (a*(A + 2*B)*cos(c + d*x)^6*(a + a*sec(c + d*x))^3*sin(c + d*x))/(14*d) + (A*cos(c + d*x)^7*(a + a*sec(c + d*x))^4*sin(c + d*x))/(8*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, (3*(4*A - 4*B + 5*C)*arctanh(sin(c + d*x)))/(8*a*d) - (2*(3*A - 4*B + 4*C)*tan(c + d*x))/(3*a*d) + (3*(4*A - 4*B + 5*C)*sec(c + d*x)*tan(c + d*x))/(8*a*d) - ((3*A - 4*B + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) + ((4*A - 4*B + 5*C)*sec(c + d*x)^3*tan(c + d*x))/(4*a*d) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, -(((2*A - 3*B + 3*C)*arctanh(sin(c + d*x)))/(2*a*d)) + (2*(3*A - 3*B + 4*C)*tan(c + d*x))/(3*a*d) - ((2*A - 3*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) + ((3*A - 3*B + 4*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, ((2*A - 2*B + 3*C)*arctanh(sin(c + d*x)))/(2*a*d) - ((A - 2*B + 2*C)*tan(c + d*x))/(a*d) + ((2*A - 2*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, ((B - C)*arctanh(sin(c + d*x)))/(a*d) + ((A - B + 2*C)*tan(c + d*x))/(a*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, (A*x)/a + (C*arctanh(sin(c + d*x)))/(a*d) - ((A - B + C)*tan(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 3, -(((A - B)*x)/a) + ((2*A - B + C)*sin(c + d*x))/(a*d) - ((A - B + C)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 4, ((3*A - 2*B + 2*C)*x)/(2*a) - ((2*A - 2*B + C)*sin(c + d*x))/(a*d) + ((3*A - 2*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 5, -(((3*A - 3*B + 2*C)*x)/(2*a)) + (2*(4*A - 3*B + 3*C)*sin(c + d*x))/(3*a*d) - ((3*A - 3*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*d) + ((4*A - 3*B + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x)), x, 6, (3*(5*A - 4*B + 4*C)*x)/(8*a) - (2*(4*A - 4*B + 3*C)*sin(c + d*x))/(3*a*d) + (3*(5*A - 4*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(8*a*d) - ((4*A - 4*B + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d) + ((5*A - 4*B + 4*C)*cos(c + d*x)^3*sin(c + d*x))/(4*a*d) - ((A - B + C)*cos(c + d*x)^3*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, -(((4*A - 7*B + 10*C)*arctanh(sin(c + d*x)))/(2*a^2*d)) + (2*(5*A - 8*B + 12*C)*tan(c + d*x))/(3*a^2*d) - ((4*A - 7*B + 10*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + ((5*A - 8*B + 12*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((4*A - 7*B + 10*C)*sec(c + d*x)^3*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, ((2*A - 4*B + 7*C)*arctanh(sin(c + d*x)))/(2*a^2*d) - (2*(2*A - 5*B + 8*C)*tan(c + d*x))/(3*a^2*d) + ((2*A - 4*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((2*A - 5*B + 8*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, ((B - 2*C)*arctanh(sin(c + d*x)))/(a^2*d) + ((A - 4*B + 10*C)*tan(c + d*x))/(3*a^2*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((B - 2*C)*sec(c + d*x)*tan(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 3, (C*arctanh(sin(c + d*x)))/(a^2*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((2*A + B - 4*C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 3, (A*x)/a^2 - ((A - B + C)*tan(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((4*A - B - 2*C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 4, -(((2*A - B)*x)/a^2) + ((10*A - 4*B + C)*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((2*A - B)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 5, ((7*A - 4*B + 2*C)*x)/(2*a^2) - (2*(8*A - 5*B + 2*C)*sin(c + d*x))/(3*a^2*d) + ((7*A - 4*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((8*A - 5*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^2, x, 6, -(((10*A - 7*B + 4*C)*x)/(2*a^2)) + (2*(12*A - 8*B + 5*C)*sin(c + d*x))/(3*a^2*d) - ((10*A - 7*B + 4*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + ((12*A - 8*B + 5*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((10*A - 7*B + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, ((2*A - 6*B + 13*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (2*(11*A - 36*B + 76*C)*tan(c + d*x))/(15*a^3*d) + ((2*A - 6*B + 13*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((A - 6*B + 11*C)*sec(c + d*x)^3*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((11*A - 36*B + 76*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, ((B - 3*C)*arctanh(sin(c + d*x)))/(a^3*d) + (2*(A - 11*B + 36*C)*tan(c + d*x))/(15*a^3*d) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((A + 4*B - 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((B - 3*C)*sec(c + d*x)*tan(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, (C*arctanh(sin(c + d*x)))/(a^3*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((3*A + 2*B - 7*C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A + 2*B - 22*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 3, -(((A - B + C)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3)) + ((A - C)*tan(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) + ((2*A + 3*B + 7*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 4, (A*x)/a^3 - ((A - B + C)*tan(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((7*A - 2*B - 3*C)*tan(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((22*A - 2*B - 3*C)*tan(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 5, -(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*sin(c + d*x))/(15*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((9*A - 4*B - C)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((3*A - B)*sin(c + d*x))/(a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 6, ((13*A - 6*B + 2*C)*x)/(2*a^3) - (2*(76*A - 36*B + 11*C)*sin(c + d*x))/(15*a^3*d) + ((13*A - 6*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((11*A - 6*B + C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((76*A - 36*B + 11*C)*cos(c + d*x)*sin(c + d*x))/(15*a^2*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^3, x, 7, -(((23*A - 13*B + 6*C)*x)/(2*a^3)) + (8*(34*A - 19*B + 9*C)*sin(c + d*x))/(15*a^3*d) - ((23*A - 13*B + 6*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (4*(34*A - 19*B + 9*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((13*A - 8*B + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((23*A - 13*B + 6*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d*(a + a*sec(c + d*x)))],


[sec(c + d*x)^5*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 7, ((2*A - 8*B + 21*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - (8*(20*A - 83*B + 216*C)*tan(c + d*x))/(105*a^4*d) + ((2*A - 8*B + 21*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - ((A - B + C)*sec(c + d*x)^5*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((B - 2*C)*sec(c + d*x)^4*tan(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - ((10*A - 52*B + 129*C)*sec(c + d*x)^3*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*(20*A - 83*B + 216*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 6, ((B - 4*C)*arctanh(sin(c + d*x)))/(a^4*d) + (2*(3*A - 80*B + 332*C)*tan(c + d*x))/(105*a^4*d) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((2*A + 5*B - 12*C)*sec(c + d*x)^3*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((3*A + 25*B - 88*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + ((B - 4*C)*sec(c + d*x)*tan(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 5, (C*arctanh(sin(c + d*x)))/(a^4*d) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) + ((4*A + 3*B - 10*C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((8*A + 6*B - 55*C)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (2*(4*A + 3*B - 80*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 4, -(((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) - ((6*A + B - 8*C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((13*A + 8*B + 6*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + (2*(13*A + 8*B + 6*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 4, -(((A - B + C)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4)) + ((8*A - B - 6*C)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) + ((6*A + 8*B + 13*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) + ((6*A + 8*B + 13*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 5, (A*x)/a^4 - ((A - B + C)*tan(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((10*A - 3*B - 4*C)*tan(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((55*A - 6*B - 8*C)*tan(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (2*(80*A - 3*B - 4*C)*tan(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 6, -(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*sin(c + d*x))/(105*a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((12*A - 5*B - 2*C)*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((88*A - 25*B - 3*C)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - ((4*A - B)*sin(c + d*x))/(a^3*d*(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^4, x, 7, ((21*A - 8*B + 2*C)*x)/(2*a^4) - (8*(216*A - 83*B + 20*C)*sin(c + d*x))/(105*a^4*d) + ((21*A - 8*B + 2*C)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((2*A - B)*cos(c + d*x)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^3) - ((129*A - 52*B + 10*C)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*sec(c + d*x))^2) - (4*(216*A - 83*B + 20*C)*cos(c + d*x)*sin(c + d*x))/(105*a^3*d*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (16*a*(99*A + 88*B + 80*C)*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (4*a*(99*A + 88*B + 80*C)*sec(c + d*x)^2*tan(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a*(99*A + 88*B + 80*C)*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a*(11*B + C)*sec(c + d*x)^4*tan(c + d*x))/(99*d*sqrt(a + a*sec(c + d*x))) + (16*(99*A + 88*B + 80*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3465*d) + (2*C*sec(c + d*x)^4*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(11*d)],
[sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (8*a*(21*A + 18*B + 16*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(21*A + 18*B + 16*C)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(9*B + C)*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (8*(21*A + 18*B + 16*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*C*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (2*a*(35*A + 28*B + 24*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(7*B + C)*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*(35*A + 28*B + 24*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*C*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (2*a*(15*A + 5*B + 7*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*(5*B + C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*sec(c + d*x)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (2*sqrt(a)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a*(3*B + C)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (sqrt(a)*(A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a*(A - 2*C)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (2*C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (sqrt(a)*(3*A + 4*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a*(A + 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (sqrt(a)*(5*A + 6*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a*(5*A + 6*B + 8*C)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*(A + 6*B)*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (sqrt(a)*(35*A + 40*B + 48*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a*(35*A + 40*B + 48*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a*(35*A + 40*B + 48*C)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*(A + 8*B)*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (8*a^2*(429*A + 374*B + 336*C)*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(429*A + 374*B + 336*C)*sec(c + d*x)^2*tan(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(99*A + 110*B + 84*C)*sec(c + d*x)^3*tan(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (8*a*(429*A + 374*B + 336*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3465*d) + (2*a*(11*B + 3*C)*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a^2*(189*A + 156*B + 136*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(63*A + 72*B + 52*C)*sec(c + d*x)^2*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(189*A + 156*B + 136*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a*(3*B + C)*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (8*a^2*(35*A + 21*B + 19*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(35*A + 21*B + 19*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*(7*B + 3*C)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*C*sec(c + d*x)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a^(3/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^2*(15*A + 20*B + 12*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*(5*B + 3*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a^(3/2)*(3*A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (a^2*(3*A - 6*B - 8*C)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*a*(B + C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d + (2*C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a^(3/2)*(7*A + 12*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^2*(5*A + 4*B - 8*C)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) - (a*(A - 4*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a^(3/2)*(11*A + 14*B + 24*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^2*(19*A + 30*B + 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*(A + 2*B)*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^(3/2)*(75*A + 88*B + 112*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^2*(75*A + 88*B + 112*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(39*A + 56*B + 48*C)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*(3*A + 8*B)*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^(3/2)*(133*A + 150*B + 176*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(128*d) + (a^2*(133*A + 150*B + 176*C)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^2*(133*A + 150*B + 176*C)*cos(c + d*x)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(67*A + 90*B + 80*C)*cos(c + d*x)^2*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (a*(3*A + 10*B)*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],


[sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (8*a^3*(10439*A + 9230*B + 8368*C)*tan(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(10439*A + 9230*B + 8368*C)*sec(c + d*x)^2*tan(c + d*x))/(15015*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2717*A + 2522*B + 2224*C)*sec(c + d*x)^3*tan(c + d*x))/(9009*d*sqrt(a + a*sec(c + d*x))) + (8*a^2*(10439*A + 9230*B + 8368*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*sec(c + d*x)^3*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(1287*d) + (2*a*(13*B + 5*C)*sec(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(143*d) + (2*C*sec(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(13*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (2*a^3*(3795*A + 3212*B + 2840*C)*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(1485*A + 1364*B + 1160*C)*sec(c + d*x)^2*tan(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(3795*A + 3212*B + 2840*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(3465*d) + (2*a^2*(33*A + 44*B + 32*C)*sec(c + d*x)^2*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(231*d) + (2*a*(11*B + 5*C)*sec(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (64*a^3*(21*A + 15*B + 13*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(21*A + 15*B + 13*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(105*d) + (2*(9*B + 5*C)*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(63*d) + (2*C*sec(c + d*x)*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^0*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (2*a^(5/2)*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d + (2*a^3*(245*A + 224*B + 160*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(35*A + 56*B + 40*C)*sqrt(a + a*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*a*(7*B + 5*C)*(a + a*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*C*(a + a*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[cos(c + d*x)^1*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(5*A + 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/d - (a^3*(15*A + 70*B + 64*C)*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(5*A + 10*B + 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(B + C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(19*A + 20*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*d) + (a^3*(27*A - 12*B - 56*C)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) - (a^2*(A - 4*B - 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d) - (a*(3*A - 4*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d) + (A*cos(c + d*x)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(25*A + 38*B + 40*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*d) + (a^3*(49*A + 54*B - 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) - (a^2*(3*A + 2*B - 8*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*(5*A + 6*B)*cos(c + d*x)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^2*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^(5/2)*(163*A + 200*B + 304*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*d) + (a^3*(299*A + 392*B + 432*C)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(17*A + 24*B + 16*C)*cos(c + d*x)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d) + (a*(5*A + 8*B)*cos(c + d*x)^2*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^(5/2)*(283*A + 326*B + 400*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(128*d) + (a^3*(283*A + 326*B + 400*C)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(787*A + 950*B + 1040*C)*cos(c + d*x)*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(79*A + 110*B + 80*C)*cos(c + d*x)^2*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d) + (a*(A + 2*B)*cos(c + d*x)^3*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^4*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (a^(5/2)*(1015*A + 1132*B + 1304*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(512*d) + (a^3*(1015*A + 1132*B + 1304*C)*sin(c + d*x))/(512*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1015*A + 1132*B + 1304*C)*cos(c + d*x)*sin(c + d*x))/(768*d*sqrt(a + a*sec(c + d*x))) + (a^3*(545*A + 628*B + 680*C)*cos(c + d*x)^2*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(115*A + 156*B + 120*C)*cos(c + d*x)^3*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(480*d) + (a*(5*A + 12*B)*cos(c + d*x)^4*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(60*d) + (A*cos(c + d*x)^5*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 6, -((sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(273*A - 129*B + 257*C)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) - (2*(21*A - 93*B + 29*C)*sec(c + d*x)*tan(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*(21*A - 3*B + 19*C)*sec(c + d*x)^2*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*(9*B - C)*sec(c + d*x)^3*tan(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^4*tan(c + d*x))/(9*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*(35*A - 91*B + 43*C)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*(35*A - 7*B + 31*C)*sec(c + d*x)*tan(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*(7*B - C)*sec(c + d*x)^2*tan(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^3*tan(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 4, -((sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d)) + (2*(15*A - 5*B + 13*C)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*(5*B - C)*sec(c + d*x)*tan(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)^2*tan(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 3, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*(3*B - C)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*C*sec(c + d*x)*tan(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + (2*C*tan(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, -(((A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(sqrt(a)*d)) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - (2*C*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + ((A + 2*C)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 5, ((7*A - 4*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - ((A - 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 6, -(((9*A - 14*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*sqrt(a)*d)) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) + ((7*A - 2*B + 8*C)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) - ((A - 6*B)*cos(c + d*x)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^2*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + a*sec(c + d*x)), x, 7, ((107*A - 72*B + 112*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(64*sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(sqrt(a)*d) - ((21*A - 56*B + 16*C)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + ((43*A - 8*B + 48*C)*cos(c + d*x)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) - ((A - 8*B)*cos(c + d*x)^2*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (A*cos(c + d*x)^3*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 6, ((11*A - 15*B + 19*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((665*A - 1029*B + 1201*C)*tan(c + d*x))/(210*a*d*sqrt(a + a*sec(c + d*x))) + ((245*A - 273*B + 397*C)*sec(c + d*x)*tan(c + d*x))/(210*a*d*sqrt(a + a*sec(c + d*x))) - ((35*A - 63*B + 67*C)*sec(c + d*x)^2*tan(c + d*x))/(70*a*d*sqrt(a + a*sec(c + d*x))) + ((7*A - 7*B + 11*C)*sec(c + d*x)^3*tan(c + d*x))/(14*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 5, -(((7*A - 11*B + 15*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((75*A - 95*B + 147*C)*tan(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) - ((15*A - 35*B + 39*C)*sec(c + d*x)*tan(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) + ((5*A - 5*B + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 4, ((3*A - 7*B + 11*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((3*A - 15*B + 19*C)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x))) + ((3*A - 3*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 3, ((A + 3*B - 7*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((A - B + 5*C)*tan(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 4, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d) - ((5*A - B - 3*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*tan(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 5, -(((3*A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(3/2)*d)) + ((9*A - 5*B + C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((3*A - B + C)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 6, ((19*A - 12*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(3/2)*d) - ((13*A - 9*B + 5*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((7*A - 6*B + 2*C)*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + ((2*A - B + C)*cos(c + d*x)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(3/2), x, 7, -(((47*A - 38*B + 24*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(8*a^(3/2)*d)) + ((17*A - 13*B + 9*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((21*A - 14*B + 12*C)*sin(c + d*x))/(8*a*d*sqrt(a + a*sec(c + d*x))) - ((13*A - 12*B + 6*C)*cos(c + d*x)*sin(c + d*x))/(12*a*d*sqrt(a + a*sec(c + d*x))) + ((5*A - 3*B + 3*C)*cos(c + d*x)^2*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 6, -(((75*A - 163*B + 283*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - ((A - B + C)*sec(c + d*x)^4*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((5*A - 13*B + 21*C)*sec(c + d*x)^3*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((735*A - 1495*B + 2671*C)*tan(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) - ((195*A - 475*B + 787*C)*sec(c + d*x)*tan(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) + ((45*A - 85*B + 157*C)*sec(c + d*x)^2*tan(c + d*x))/(80*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 5, ((19*A - 75*B + 163*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sec(c + d*x)^3*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((A - 9*B + 17*C)*sec(c + d*x)^2*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((27*A - 147*B + 299*C)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x))) + ((15*A - 39*B + 95*C)*sec(c + d*x)*tan(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 4, ((5*A + 19*B - 75*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((3*A + 5*B - 13*C)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((A - 9*B + 49*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 3, ((3*A + 5*B + 19*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((7*A + B - 9*C)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 5, (2*A*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d) - ((43*A - 3*B - 5*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*tan(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((11*A - 3*B - 5*C)*tan(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 6, -(((5*A - 2*B)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(a^(5/2)*d)) + ((115*A - 43*B + 3*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((15*A - 7*B - C)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((35*A - 11*B + 3*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(5/2), x, 7, ((39*A - 20*B + 8*C)*arctan((sqrt(a)*tan(c + d*x))/sqrt(a + a*sec(c + d*x))))/(4*a^(5/2)*d) - ((219*A - 115*B + 43*C)*arctan((sqrt(a)*tan(c + d*x))/(sqrt(2)*sqrt(a + a*sec(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*cos(c + d*x)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((19*A - 11*B + 3*C)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((63*A - 35*B + 11*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + ((31*A - 15*B + 7*C)*cos(c + d*x)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, -((2*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(5*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*C*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*sec(c + d*x)^(1/2), x, 6, -((2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(1/2), x, 5, (2*(A - C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*C*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(3/2), x, 5, (2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(5/2), x, 6, (2*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sec(c + d*x)^(7/2), x, 7, (6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(5*A + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (-2*a*(5*A + 3*B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(7*A + 7*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(5*A + 3*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(7*A + 7*B + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*(7*B + 2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (-2*a*(5*A + 5*B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(3*A + B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(5*A + 5*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(5*B + 2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 6, (2*a*(A - B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(3*A + 3*B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(3*B + 2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 6, (2*a*(A + B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*(A + 3*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a*(A - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 6, (2*a*(3*A + 5*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(A + B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(2*A + 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 7, (2*a*(3*A + 3*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(5*A + 7*B + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(2*A + 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a*(5*A + 7*B + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (2*a*(7*A + 9*B + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(5*A + 5*B + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(2*A + 9*B)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a*(7*A + 9*B + 9*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*a*(5*A + 5*B + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (-4*a^2*(12*A + 9*B + 8*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(12*A + 9*B + 8*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (2*a*(9*B + 4*C)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (-4*a^2*(5*A + 4*B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(5*A + 4*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(35*A + 49*B + 33*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(7*B + 4*C)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 7, (-4*a^2*(5*B + 4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(3*A + 2*B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(5*B + 4*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 7, (4*a^2*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (4*a^2*(2*A + 3*B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(A - 3*B - 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) - (2*a*(A - C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 7, (4*a^2*(4*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(A + 2*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(7*A + 5*B - 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(4*A + 5*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 7, (4*a^2*(3*A + 4*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a*(4*A + 7*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (4*a^2*(8*A + 9*B + 12*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (4*a^2*(5*A + 6*B + 7*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*a*(4*A + 9*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 9, (4*a^2*(7*A + 8*B + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*a^2*(89*A + 121*B + 99*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (4*a^2*(7*A + 8*B + 9*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (4*a^2*(50*A + 55*B + 66*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a*(4*A + 11*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 10, (-4*a^3*(21*A + 17*B + 15*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(21*A + 17*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(1155*d) + (2*a^2*(99*A + 143*B + 105*C)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(693*d) + (2*a*(11*B + 6*C)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d) + (2*C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d)],
[sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (-4*a^3*(27*A + 21*B + 17*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(27*A + 21*B + 17*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(42*A + 41*B + 32*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(63*A + 99*B + 73*C)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(3*B + 2*C)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d)],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 8, (-4*a^3*(5*A + 9*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(5*A + 9*B + 7*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(7*B + 6*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 8, (4*a^3*(5*A - 5*B - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*(5*A + 20*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(5*A - 5*B - 9*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a*(5*A - 3*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 8, (4*a^3*(9*A + 5*B - 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(3*A + 5*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(6*A - 5*B - 20*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(9*A + 5*B - 5*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(6*A + 5*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 8, (4*a^3*(7*A + 9*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) - (4*a^3*(41*A + 42*B - 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 9*B + 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*(6*A + 7*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 8, (4*a^3*(17*A + 21*B + 27*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(73*A + 99*B + 63*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*a*(2*A + 3*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 9, (4*a^3*(15*A + 17*B + 21*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*sin(c + d*x))/(1155*d*sec(c + d*x)^(3/2)) + (4*a^3*(105*A + 121*B + 143*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(105*A + 143*B + 99*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*a*(6*A + 11*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(13/2), x, 10, (4*a^3*(175*A + 195*B + 221*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (4*a^3*(95*A + 105*B + 121*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (20*a^3*(236*A + 273*B + 286*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (4*a^3*(175*A + 195*B + 221*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (4*a^3*(95*A + 105*B + 121*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*a^2*(145*A + 195*B + 143*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(7/2)) + (2*a*(6*A + 13*B)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(9/2)) + (2*A*(a + a*sec(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 8, (-3*(5*A - 5*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a*d) - ((3*A - 5*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + (3*(5*A - 5*B + 7*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d) - ((3*A - 5*B + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) + ((5*A - 5*B + 7*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 7, ((A - 3*B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) + ((3*A - 3*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) - ((A - 3*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + ((3*A - 3*B + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 6, -(((A - B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d)) + ((A + B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) + ((A - B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))), x, 5, ((3*A - B + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, -(((3*A - 3*B + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d)) + ((5*A - 3*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((5*A - 3*B + 3*C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (3*(7*A - 5*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a*d) - ((5*A - 5*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + ((7*A - 5*B + 5*C)*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - ((5*A - 5*B + 3*C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 8, ((A - 4*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((2*A - 5*B + 10*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - 4*B + 7*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + ((2*A - 5*B + 10*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((A - 4*B + 7*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 7, ((B - 4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((A + 2*B - 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((B - 4*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A + 2*B - 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 6, -(((A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + ((2*A + B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) + ((A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2), x, 6, ((4*A - B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) - ((5*A - 2*B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((5*A - 2*B - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 7, -(((7*A - 4*B + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + ((10*A - 5*B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((10*A - 5*B + 2*C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - ((7*A - 4*B + C)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 8, ((56*A - 35*B + 20*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*(3*A - 2*B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((56*A - 35*B + 20*C)*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*(3*A - 2*B + C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((3*A - 2*B + C)*sin(c + d*x))/(a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 9, ((9*A - 49*B + 119*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A - 13*B + 33*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((9*A - 49*B + 119*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((3*A - 13*B + 33*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^3*d) - ((A - B + C)*sec(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((B - 2*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - ((9*A - 49*B + 119*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 8, ((A + 9*B - 49*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*B - 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + 9*B - 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((2*A + 3*B - 8*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A + 3*B - 13*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((A - B - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((4*A + B - 6*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((A - B - 9*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((9*A + B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A + B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + ((6*A - B - 4*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + ((3*A + B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3), x, 7, ((49*A - 9*B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 3*B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((8*A - 3*B - 2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((13*A - 3*B - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 8, -((119*A - 49*B + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((33*A - 13*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((33*A - 13*B + 3*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - ((2*A - B)*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - ((119*A - 49*B + 9*C)*sin(c + d*x))/(30*a^2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 9, (7*(33*A - 17*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((63*A - 33*B + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) - ((63*A - 33*B + 13*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) - ((12*A - 7*B + 2*C)*sin(c + d*x))/(15*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((63*A - 33*B + 13*C)*sin(c + d*x))/(10*a^2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a*(48*A + 40*B + 35*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 40*B + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 40*B + 35*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*(8*B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a*(8*A + 6*B + 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(8*A + 6*B + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + (a*(6*B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (a*(8*A + 4*B + 3*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a*(4*B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d)],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 4, (sqrt(2)*a*(2*B + C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a*(2*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x))) + (C*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/d],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 4, (2*sqrt(2)*a*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 3, (2*a*(A + 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(8*A + 10*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 4, (2*a*(A + 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(24*A + 28*B + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a*(24*A + 28*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (2*a*(A + 9*B)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a*(16*A + 18*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (a^2*(176*A + 150*B + 133*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 150*B + 133*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 150*B + 133*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 90*B + 67*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(240*d*sqrt(a + a*sec(c + d*x))) + (a*(10*B + 3*C)*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d) + (C*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^2*(112*A + 88*B + 75*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(112*A + 88*B + 75*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*d*sqrt(a + a*sec(c + d*x))) + (a^2*(48*A + 56*B + 39*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(96*d*sqrt(a + a*sec(c + d*x))) + (a*(8*B + 3*C)*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^2*(24*A + 14*B + 11*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 30*B + 19*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a*(2*B + C)*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 5, (a^2*(8*A + 12*B + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 4*B - 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (a*(4*B + 3*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 5, (sqrt(2)*a^2*(2*B + 3*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 6*B - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x))) - (a*(2*A - 3*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 5, (2*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(12*A + 20*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(19*A + 21*B + 35*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*(3*A + 7*B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (2*a^2*(52*A + 72*B + 63*C)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 156*B + 189*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^2*(136*A + 156*B + 189*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 3*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 6, (2*a^2*(84*A + 110*B + 99*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(1155*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(336*A + 374*B + 429*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 11*B)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


[sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (a^3*(1304*A + 1132*B + 1015*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(256*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1132*B + 1015*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(512*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1132*B + 1015*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(768*d*sqrt(a + a*sec(c + d*x))) + (a^3*(680*A + 628*B + 545*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(120*A + 156*B + 115*C)*sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(480*d) + (a*(12*B + 5*C)*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(60*d) + (C*sec(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],
[sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (a^3*(400*A + 326*B + 283*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(400*A + 326*B + 283*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(128*d*sqrt(a + a*sec(c + d*x))) + (a^3*(1040*A + 950*B + 787*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(960*d*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 110*B + 79*C)*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d) + (a*(2*B + C)*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^3*(304*A + 200*B + 163*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(432*A + 392*B + 299*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(192*d*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 24*B + 17*C)*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d) + (a*(8*B + 5*C)*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(1/2), x, 6, (a^3*(40*A + 38*B + 25*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(24*A - 54*B - 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 42*B + 31*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d) + (a*(6*B + 5*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (C*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 6, (a^3*(8*A + 20*B + 19*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + (a^3*(56*A + 12*B - 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) - (a^2*(8*A - 12*B - 21*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(12*d) - (a*(4*A - 3*C)*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 6, (sqrt(2)*a^3*(2*B + 5*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (a^3*(64*A + 70*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) - (a^2*(16*A + 10*B - 15*C)*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(A + B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 6, (2*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(160*A + 224*B + 245*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(40*A + 56*B + 35*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a*(5*A + 7*B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (16*a^2*(13*A + 15*B + 21*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*a*(13*A + 15*B + 21*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (2*(5*A + 9*B)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 6, (2*a^3*(1160*A + 1364*B + 1485*C)*sin(c + d*x))/(3465*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2840*A + 3212*B + 3795*C)*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (4*a^3*(2840*A + 3212*B + 3795*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(32*A + 44*B + 33*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)) + (2*a*(5*A + 11*B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(13/2), x, 7, (2*a^3*(2224*A + 2522*B + 2717*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (2*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(15015*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (8*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(45045*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^3*(8368*A + 9230*B + 10439*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 182*B + 143*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(7/2)) + (2*a*(5*A + 13*B)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(143*d*sec(c + d*x)^(9/2)) + (2*A*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(13*d*sec(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 8, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) - ((8*A - 14*B + 9*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + ((8*A - 2*B + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(8*d*sqrt(a + a*sec(c + d*x))) + ((6*B - C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(12*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 7, (-2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + ((8*A - 4*B + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*sec(c + d*x))) + ((4*B - C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (C*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x))), x, 6, (-2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 4, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 5, (-2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - (2*(A - 5*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(13*A - 5*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - (2*(A - 7*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (2*(31*A - 7*B + 35*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A - 91*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x)))],

[(sqrt(sec(c + d*x))*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(a*A - A*b - a*B + b*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*A*b + 2*a*B - b*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(d*sqrt(a + a*sec(c + d*x))) + (b*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(d*sqrt(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 8, -(((5*A - 9*B + 13*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) + ((8*A - 12*B + 19*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((2*A - 6*B + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a*d*sqrt(a + a*sec(c + d*x))) + ((A - B + 2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 7, ((A - 5*B + 9*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - 3*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((A - B + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 6, ((3*A + B - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2)), x, 4, -(((7*A - 3*B - C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x)))) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((5*A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 5, ((11*A - 7*B + 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((7*A - 3*B + 3*C)*sin(c + d*x))/(6*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((19*A - 15*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, -((15*A - 11*B + 7*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(2*a*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((9*A - 5*B + 5*C)*sin(c + d*x))/(10*a*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((39*A - 35*B + 15*C)*sin(c + d*x))/(30*a*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((147*A - 95*B + 75*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x)))],


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 8, ((3*A - 43*B + 115*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((A + 7*B - 15*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((3*A - 11*B + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 7, ((5*A + 3*B - 43*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B - 11*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[(sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 4, ((19*A + 5*B + 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x))*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((9*A - B - 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2)), x, 5, -((75*A - 19*B - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 5*B - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((49*A - 9*B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 6, ((163*A - 75*B + 19*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((17*A - 9*B + C)*sin(c + d*x))/(16*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((95*A - 39*B + 15*C)*sin(c + d*x))/(48*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((299*A - 147*B + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, -((283*A - 163*B + 75*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2)*sqrt(sec(c + d*x)))/(16*a^2*d*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - ((21*A - 13*B + 5*C)*sin(c + d*x))/(16*a*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)) + ((157*A - 85*B + 45*C)*sin(c + d*x))/(80*a^2*d*sec(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) - ((787*A - 475*B + 195*C)*sin(c + d*x))/(240*a^2*d*sqrt(sec(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((2671*A - 1495*B + 735*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/3)
#


[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(2/3), x, 1, Int((a + a*sec(c + d*x))^(2/3)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x)],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + a*sec(c + d*x))^(1/3), x, 1, Int((a + a*sec(c + d*x))^(1/3)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x)],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(1/3), x, 0, Int((A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(1/3), x)],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(2/3), x, 0, Int((A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + a*sec(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^n


# ::Subsubsection::Closed:: 
#n>=0


[cos(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (6*B*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(5*A + 7*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(5*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*B*EllipticF((c + d*x)/2, 2))/(3*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (2*B*EllipticE((c + d*x)/2, 2))/d + (2*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (2*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*B*EllipticF((c + d*x)/2, 2))/d + (2*C*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(cos(c + d*x)), x, 5, (-2*B*EllipticE((c + d*x)/2, 2))/d + (2*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*C*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(3/2), x, 6, (-2*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*B*EllipticF((c + d*x)/2, 2))/(3*d) + (2*C*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(5*A + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(5/2), x, 7, (-6*B*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*C*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*B*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(7*A + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (6*B*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (2*a*(7*A + 9*B + 9*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*a*(5*A + 5*B + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(5*A + 5*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(7*A + 9*B + 9*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a*(2*A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (2*a*(3*A + 3*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(5*A + 7*B + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(5*A + 7*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(2*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a*(3*A + 5*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(A + B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(2*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a*(A + B - C)*EllipticE((c + d*x)/2, 2))/d + (2*a*(A + 3*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a*(A - 3*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*a*(A - B - C)*EllipticE((c + d*x)/2, 2))/d + (2*a*(3*A + 3*B + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(3*B + 2*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (-2*a*(5*A + 5*B + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(3*A + B + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(5*B + 2*C)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 5*B + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (-2*a*(5*A + 3*B + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(7*A + 7*B + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(7*B + 2*C)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*(7*A + 7*B + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 3*B + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (4*a^2*(7*A + 8*B + 9*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^2*(50*A + 55*B + 66*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^2*(7*A + 8*B + 9*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(89*A + 121*B + 99*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*a*(4*A + 11*B)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^2*(8*A + 9*B + 12*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^2*(5*A + 6*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(4*A + 9*B)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (4*a^2*(3*A + 4*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(4*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (4*a^2*(4*A + 5*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(A + 2*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(7*A + 5*B - 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(4*A + 5*B)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (4*a^2*(A - C)*EllipticE((c + d*x)/2, 2))/d + (4*a^2*(2*A + 3*B + 2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(A - 3*B - 5*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) - (2*a*(A - C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (-4*a^2*(5*B + 4*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(5*B + 4*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (-4*a^2*(5*A + 4*B + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^2*(5*A + 4*B + 3*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*(7*B + 4*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(3/2)) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 8, (-4*a^2*(12*A + 9*B + 8*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^2*(7*A + 6*B + 5*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^2*(12*A + 9*B + 8*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(9*B + 4*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(5/2)) + (2*C*(a + a*sec(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (4*a^3*(175*A + 195*B + 221*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (4*a^3*(95*A + 105*B + 121*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(95*A + 105*B + 121*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(175*A + 195*B + 221*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(585*d) + (20*a^3*(236*A + 273*B + 286*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d) + (2*a^2*(145*A + 195*B + 143*C)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d) + (2*a*(6*A + 13*B)*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (4*a^3*(15*A + 17*B + 21*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(105*A + 121*B + 143*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d) + (2*a^2*(105*A + 143*B + 99*C)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(693*d) + (2*a*(6*A + 11*B)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^3*(17*A + 21*B + 27*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(73*A + 99*B + 63*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(2*A + 3*B)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^3*(7*A + 9*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (4*a^3*(41*A + 42*B - 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(7*A + 9*B + 5*C)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(6*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^3*(9*A + 5*B - 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(3*A + 5*B + 5*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (4*a^3*(6*A - 5*B - 20*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(9*A + 5*B - 5*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(6*A + 5*B)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (4*a^3*(5*A - 5*B - 9*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a^3*(5*A + 20*B + 21*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(5*A - 5*B - 9*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a*(5*A - 3*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (-4*a^3*(5*A + 9*B + 7*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(5*A + 9*B + 7*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(7*B + 6*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(7*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 8, (-4*a^3*(27*A + 21*B + 17*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(27*A + 21*B + 17*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(63*A + 99*B + 73*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*a*(3*B + 2*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 9, (-4*a^3*(21*A + 17*B + 15*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*sin(c + d*x))/(1155*d*cos(c + d*x)^(5/2)) + (4*a^3*(143*A + 121*B + 105*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (4*a^3*(21*A + 17*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(99*A + 143*B + 105*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(693*d*cos(c + d*x)^(5/2)) + (2*a*(11*B + 6*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d*cos(c + d*x)^(5/2)) + (2*C*(a + a*sec(c + d*x))^3*sin(c + d*x))/(11*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(15/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 10, (8*a^4*(168*A + 185*B + 208*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (8*a^4*(91*A + 100*B + 113*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (8*a^4*(91*A + 100*B + 113*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (8*a^4*(168*A + 185*B + 208*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(585*d) + (4*a^4*(2289*A + 2545*B + 2795*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d) + (4*a^3*(147*A + 170*B + 169*C)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d) + (2*a^2*(217*A + 285*B + 195*C)*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(2145*d) + (2*a*(8*A + 15*B)*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(195*d) + (2*A*cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(15*d)],
[cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (8*a^4*(185*A + 208*B + 247*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (8*a^4*(100*A + 113*B + 132*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (8*a^4*(100*A + 113*B + 132*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^4*(5255*A + 6019*B + 6721*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15015*d) + (4*a^3*(1355*A + 1612*B + 1573*C)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(9009*d) + (2*a^2*(13*A + 17*B + 11*C)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d) + (2*a*(8*A + 13*B)*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (8*a^4*(16*A + 19*B + 24*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (8*a^4*(113*A + 132*B + 187*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^4*(667*A + 803*B + 913*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(1155*d) + (4*a^3*(769*A + 946*B + 891*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))*sin(c + d*x))/(3465*d) + (2*a^2*(43*A + 55*B + 33*C)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(231*d) + (2*a*(8*A + 11*B)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (8*a^4*(19*A + 24*B + 21*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a^4*(12*A + 17*B + 28*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (4*a^4*(73*A + 83*B + 7*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (4*a^3*(19*A + 24*B + 21*C)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))*sin(c + d*x))/(45*d) + (2*a^2*(97*A + 117*B + 63*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(315*d) + (2*a*(8*A + 9*B)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (8*a^4*(8*A + 7*B)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^4*(17*A + 28*B + 35*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (4*a^4*(83*A + 7*B - 175*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (4*a^3*(8*A + 7*B)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(73*A + 77*B + 35*C)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2*sin(c + d*x))/(105*d) + (2*a*(8*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (56*a^4*(A - C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^4*(4*A + 5*B + 4*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (4*a^4*(A - 25*B - 41*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (28*a^3*(A - C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(11*A + 5*B - 3*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(8*A + 5*B)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (-8*a^4*(7*B + 8*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^4*(35*A + 28*B + 17*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^4*(175*A + 287*B + 253*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (4*a^3*(7*B + 8*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*a^2*(35*A - 21*B - 39*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (2*a*(7*A - 3*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^4*sin(c + d*x))/(3*d)],
[sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (-8*a^4*(21*A + 24*B + 19*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a^4*(28*A + 17*B + 12*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^4*(287*A + 253*B + 193*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (4*a^3*(21*A + 24*B + 19*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(45*d*sqrt(cos(c + d*x))) + (2*a^2*(63*A + 117*B + 97*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (2*a*(9*B + 8*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(63*d*sqrt(cos(c + d*x))) + (2*C*(a + a*sec(c + d*x))^4*sin(c + d*x))/(9*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(cos(c + d*x)), x, 9, (-8*a^4*(24*A + 19*B + 16*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a^4*(187*A + 132*B + 113*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^4*(913*A + 803*B + 667*C)*sin(c + d*x))/(1155*d*cos(c + d*x)^(3/2)) + (8*a^4*(24*A + 19*B + 16*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (4*a^3*(891*A + 946*B + 769*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)) + (2*a^2*(33*A + 55*B + 43*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (2*a*(11*B + 8*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(99*d*cos(c + d*x)^(3/2)) + (2*C*(a + a*sec(c + d*x))^4*sin(c + d*x))/(11*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 10, (-8*a^4*(247*A + 208*B + 185*C)*EllipticE((c + d*x)/2, 2))/(195*d) + (8*a^4*(132*A + 113*B + 100*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^4*(6721*A + 6019*B + 5255*C)*sin(c + d*x))/(15015*d*cos(c + d*x)^(5/2)) + (8*a^4*(132*A + 113*B + 100*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (8*a^4*(247*A + 208*B + 185*C)*sin(c + d*x))/(195*d*sqrt(cos(c + d*x))) + (4*a^3*(1573*A + 1612*B + 1355*C)*(a + a*sec(c + d*x))*sin(c + d*x))/(9009*d*cos(c + d*x)^(5/2)) + (2*a^2*(11*A + 17*B + 13*C)*(a + a*sec(c + d*x))^2*sin(c + d*x))/(99*d*cos(c + d*x)^(5/2)) + (2*a*(13*B + 8*C)*(a + a*sec(c + d*x))^3*sin(c + d*x))/(143*d*cos(c + d*x)^(5/2)) + (2*C*(a + a*sec(c + d*x))^4*sin(c + d*x))/(13*d*cos(c + d*x)^(5/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 7, (-3*(7*A - 7*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) + (5*(9*A - 7*B + 7*C)*EllipticF((c + d*x)/2, 2))/(21*a*d) + (5*(9*A - 7*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*a*d) - ((7*A - 7*B + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) + ((9*A - 7*B + 7*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*a*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 6, (3*(7*A - 5*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) - ((5*A - 5*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) - ((5*A - 5*B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + ((7*A - 5*B + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 5, -(((3*A - 3*B + C)*EllipticE((c + d*x)/2, 2))/(a*d)) + ((5*A - 3*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((5*A - 3*B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x)), x, 4, ((3*A - B + C)*EllipticE((c + d*x)/2, 2))/(a*d) - ((A - B - C)*EllipticF((c + d*x)/2, 2))/(a*d) - ((A - B + C)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))), x, 5, -(((A - B + 3*C)*EllipticE((c + d*x)/2, 2))/(a*d)) + ((A + B - C)*EllipticF((c + d*x)/2, 2))/(a*d) + ((A - B + 3*C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))), x, 6, ((A - 3*B + 3*C)*EllipticE((c + d*x)/2, 2))/(a*d) + ((3*A - 3*B + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((3*A - 3*B + 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - ((A - 3*B + 3*C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))), x, 7, (-3*(5*A - 5*B + 7*C)*EllipticE((c + d*x)/2, 2))/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a*d) + ((5*A - 5*B + 7*C)*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - ((3*A - 5*B + 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) + (3*(5*A - 5*B + 7*C)*sin(c + d*x))/(5*a*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 8, (-7*(11*A - 8*B + 5*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) + (5*(30*A - 21*B + 14*C)*EllipticF((c + d*x)/2, 2))/(21*a^2*d) + (5*(30*A - 21*B + 14*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*a^2*d) - (7*(11*A - 8*B + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) + ((30*A - 21*B + 14*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*a^2*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((11*A - 8*B + 5*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 7, ((56*A - 35*B + 20*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(3*A - 2*B + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (5*(3*A - 2*B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + ((56*A - 35*B + 20*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((3*A - 2*B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 6, -(((7*A - 4*B + C)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + ((10*A - 5*B + 2*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + ((10*A - 5*B + 2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2) - ((7*A - 4*B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^2, x, 5, ((4*A - B)*EllipticE((c + d*x)/2, 2))/(a^2*d) - ((5*A - 2*B - C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((5*A - 2*B - C)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2), x, 5, -(((A - C)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + ((2*A + B + 2*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) + ((A - C)*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2), x, 6, ((B - 4*C)*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((A + 2*B - 5*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((B - 4*C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((A + 2*B - 5*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2), x, 7, ((A - 4*B + 7*C)*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((2*A - 5*B + 10*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + ((2*A - 5*B + 10*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - ((A - 4*B + 7*C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - ((A - 4*B + 7*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2), x, 8, -((20*A - 35*B + 56*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + ((20*A - 35*B + 56*C)*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(5/2)) - (5*(A - 2*B + 3*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + ((20*A - 35*B + 56*C)*sin(c + d*x))/(5*a^2*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^2) - ((A - 2*B + 3*C)*sin(c + d*x))/(a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 8, (7*(33*A - 17*B + 7*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((63*A - 33*B + 13*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((63*A - 33*B + 13*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((12*A - 7*B + 2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - ((63*A - 33*B + 13*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^2*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 7, -((119*A - 49*B + 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((33*A - 13*B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - ((2*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*sec(c + d*x))^2) - ((119*A - 49*B + 9*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a^2*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^3, x, 6, ((49*A - 9*B - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((13*A - 3*B - C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - ((8*A - 3*B - 2*C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((13*A - 3*B - C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3), x, 6, -((9*A + B - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A + B + C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) + ((6*A - B - 4*C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((3*A + B + C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3), x, 6, -((A - B - 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + B + 3*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) + ((4*A + B - 6*C)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) + ((A - B - 9*C)*sin(c + d*x))/(10*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3), x, 7, ((A + 9*B - 49*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 3*B - 13*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + 9*B - 49*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) + ((2*A + 3*B - 8*C)*sin(c + d*x))/(15*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((A + 3*B - 13*C)*sin(c + d*x))/(6*a^2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3), x, 8, ((9*A - 49*B + 119*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A - 13*B + 33*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((3*A - 13*B + 33*C)*sin(c + d*x))/(6*a^3*d*cos(c + d*x)^(3/2)) - ((9*A - 49*B + 119*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) + ((B - 2*C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - ((9*A - 49*B + 119*C)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^4, x, 9, ((407*A - 176*B + 57*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d) - ((260*A - 113*B + 36*C)*EllipticF((1/2)*(c + d*x), 2))/(14*a^4*d) - ((260*A - 113*B + 36*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(14*a^4*d) + ((407*A - 176*B + 57*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^4*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((15*A - 8*B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((305*A - 137*B + 39*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(210*a^2*d*(a + a*sec(c + d*x))^2) - (3*(260*A - 113*B + 36*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(70*a^3*d*(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^4, x, 8, -(((176*A - 57*B + 8*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d)) + ((339*A - 108*B + 17*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) + ((339*A - 108*B + 17*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(42*a^4*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d*(a + a*sec(c + d*x))^4) - ((13*A - 6*B - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(35*a*d*(a + a*sec(c + d*x))^3) - ((43*A - 15*B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(42*a^2*d*(a + a*sec(c + d*x))^2) - ((176*A - 57*B + 8*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a^3*d*(a + a*sec(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^4, x, 7, ((57*A - 8*B - C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d) - ((108*A - 17*B - 4*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^4) - ((11*A - 4*B - 3*C)*sin(c + d*x))/(35*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) - ((141*A - 29*B - 13*C)*sin(c + d*x))/(210*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) - ((108*A - 17*B - 4*C)*sin(c + d*x))/(42*a^3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^4), x, 7, -(((8*A + B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d)) + ((17*A + 4*B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^4) + ((9*A - 2*B - 5*C)*sin(c + d*x))/(35*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^3) + ((29*A + 13*B + 15*C)*sin(c + d*x))/(210*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((17*A + 4*B + 3*C)*sin(c + d*x))/(42*a^3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^4), x, 7, -(((A - C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d)) + ((4*A + 3*B + 4*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^4) + ((A - C)*sin(c + d*x))/(5*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^3) + ((A - 15*B - 41*C)*sin(c + d*x))/(210*a^2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^2) + ((4*A + 3*B + 4*C)*sin(c + d*x))/(42*a^3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^4), x, 7, ((B + 8*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d) + ((3*A + 4*B + 17*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^4) + ((5*A + 2*B - 9*C)*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^3) + ((15*A - B - 83*C)*sin(c + d*x))/(210*a^2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2) - ((B + 8*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^4), x, 8, ((A + 8*B - 57*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d) + ((4*A + 17*B - 108*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) - ((A + 8*B - 57*C)*sin(c + d*x))/(10*a^4*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(7*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^4) + ((3*A + 4*B - 11*C)*sin(c + d*x))/(35*a*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^3) + ((13*A + 29*B - 141*C)*sin(c + d*x))/(210*a^2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^2) + ((4*A + 17*B - 108*C)*sin(c + d*x))/(42*a^3*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^4), x, 9, ((8*A - 57*B + 176*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^4*d) + ((17*A - 108*B + 339*C)*EllipticF((1/2)*(c + d*x), 2))/(42*a^4*d) + ((17*A - 108*B + 339*C)*sin(c + d*x))/(42*a^4*d*cos(c + d*x)^(3/2)) - ((8*A - 57*B + 176*C)*sin(c + d*x))/(10*a^4*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(7*d*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^4) + ((A + 6*B - 13*C)*sin(c + d*x))/(35*a*d*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^3) - ((A - 15*B + 43*C)*sin(c + d*x))/(42*a^2*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^2) - ((8*A - 57*B + 176*C)*sin(c + d*x))/(30*a^3*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Sec[c+d x]+C Sec[c+d x]^2) (a+a Sec[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(9/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (16*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a*(16*A + 18*B + 21*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(16*A + 18*B + 21*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (4*a*(24*A + 28*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(24*A + 28*B + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 3, (2*a*(8*A + 10*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (2*sqrt(2)*a*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (sqrt(2)*a*(2*B + C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(2*A - C)*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(1/2), x, 4, (a*(8*A + 4*B + 3*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(4*B + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 5, (a*(8*A + 6*B + 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(6*B + C)*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(8*A + 6*B + 5*C)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2))],
[(sqrt(a + a*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 6, (a*(48*A + 40*B + 35*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(8*B + C)*sin(c + d*x))/(24*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 40*B + 35*C)*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a*(48*A + 40*B + 35*C)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (C*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (16*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^2*(336*A + 374*B + 429*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(336*A + 374*B + 429*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(84*A + 110*B + 99*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 11*B)*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (4*a^2*(136*A + 156*B + 189*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 156*B + 189*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(52*A + 72*B + 63*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*sec(c + d*x))) + (2*a*(A + 3*B)*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(19*A + 21*B + 35*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(3*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (2*sqrt(2)*a^2*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(12*A + 20*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (sqrt(2)*a^2*(2*B + 3*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A + 6*B - 3*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a*(2*A - 3*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (a^2*(8*A + 12*B + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(8*A - 4*B - 5*C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a*(4*B + 3*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(1/2), x, 5, (a^2*(24*A + 14*B + 11*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 30*B + 19*C)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*(2*B + C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 6, (a^2*(112*A + 88*B + 75*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(48*A + 56*B + 39*C)*sin(c + d*x))/(96*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(112*A + 88*B + 75*C)*sin(c + d*x))/(64*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*(8*B + 3*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d*cos(c + d*x)^(5/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (a^2*(176*A + 150*B + 133*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 90*B + 67*C)*sin(c + d*x))/(240*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 150*B + 133*C)*sin(c + d*x))/(192*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(176*A + 150*B + 133*C)*sin(c + d*x))/(128*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a*(10*B + 3*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(40*d*cos(c + d*x)^(7/2)) + (C*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(13/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 7, (16*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(45045*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (8*a^3*(8368*A + 9230*B + 10439*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(8368*A + 9230*B + 10439*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15015*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2224*A + 2522*B + 2717*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(136*A + 182*B + 143*C)*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(1287*d) + (2*a*(5*A + 13*B)*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (4*a^3*(2840*A + 3212*B + 3795*C)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(2840*A + 3212*B + 3795*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^3*(1160*A + 1364*B + 1485*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3465*d*sqrt(a + a*sec(c + d*x))) + (2*a^2*(32*A + 44*B + 33*C)*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(231*d) + (2*a*(5*A + 11*B)*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (16*a^2*(13*A + 15*B + 21*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(13*A + 15*B + 21*C)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(105*d) + (2*(5*A + 9*B)*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (2*sqrt(2)*a^3*C*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^3*(160*A + 224*B + 245*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a^2*(40*A + 56*B + 35*C)*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(5*A + 7*B)*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (sqrt(2)*a^3*(2*B + 5*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(64*A + 70*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a^2*(16*A + 10*B - 15*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(A + B)*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^3*(8*A + 20*B + 19*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(56*A + 12*B - 27*C)*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (a^2*(8*A - 12*B - 21*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) - (a*(4*A - 3*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 6, (a^3*(40*A + 38*B + 25*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(24*A - 54*B - 49*C)*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^2*(24*A + 42*B + 31*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))) + (a*(6*B + 5*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(1/2), x, 6, (a^3*(304*A + 200*B + 163*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(432*A + 392*B + 299*C)*sin(c + d*x))/(192*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(16*A + 24*B + 17*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(32*d*cos(c + d*x)^(3/2)) + (a*(8*B + 5*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (a^3*(400*A + 326*B + 283*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(1040*A + 950*B + 787*C)*sin(c + d*x))/(960*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(400*A + 326*B + 283*C)*sin(c + d*x))/(128*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(80*A + 110*B + 79*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(240*d*cos(c + d*x)^(5/2)) + (a*(2*B + C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d*cos(c + d*x)^(5/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(c + d*x)^(5/2), x, 8, (a^3*(1304*A + 1132*B + 1015*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(256*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (a^3*(680*A + 628*B + 545*C)*sin(c + d*x))/(960*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1132*B + 1015*C)*sin(c + d*x))/(768*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + (a^3*(1304*A + 1132*B + 1015*C)*sin(c + d*x))/(512*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))) + (a^2*(120*A + 156*B + 115*C)*sqrt(a + a*sec(c + d*x))*sin(c + d*x))/(480*d*cos(c + d*x)^(7/2)) + (a*(12*B + 5*C)*(a + a*sec(c + d*x))^(3/2)*sin(c + d*x))/(60*d*cos(c + d*x)^(7/2)) + (C*(a + a*sec(c + d*x))^(5/2)*sin(c + d*x))/(6*d*cos(c + d*x)^(7/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(43*A - 91*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*(31*A - 7*B + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*sec(c + d*x))) - (2*(A - 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 5, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))) + (2*(13*A - 5*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*sec(c + d*x))) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 4, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - (2*(A - 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(1/2)*sqrt(a + a*sec(c + d*x))), x, 6, (2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x))), x, 7, (-2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((8*A - 4*B + 7*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + ((4*B - C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))), x, 8, (2*(A - B + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((8*A - 14*B + 9*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (C*sin(c + d*x))/(3*d*cos(c + d*x)^(7/2)*sqrt(a + a*sec(c + d*x))) + ((6*B - C)*sin(c + d*x))/(12*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) + ((8*A - 2*B + 7*C)*sin(c + d*x))/(8*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],

[(sqrt(cos(c + d*x))*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + a*sec(c + d*x)), x, 6, (-2*(a*A - A*b - a*B + b*B)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*b*B*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 6, -(((15*A - 11*B + 7*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) + ((147*A - 95*B + 75*C)*sin(c + d*x))/(30*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((39*A - 35*B + 15*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*sec(c + d*x))) + ((9*A - 5*B + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 5, ((11*A - 7*B + 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*sec(c + d*x))^(3/2)) - ((19*A - 15*B + 3*C)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((7*A - 3*B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(3/2), x, 4, -(((7*A - 3*B - C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))) - ((A - B + C)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((5*A - B + C)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(3/2)), x, 6, ((3*A + B - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2)), x, 7, ((A - 5*B + 9*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - 3*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + ((A - B + 3*C)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)), x, 8, -((5*A - 9*B + 13*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((8*A - 12*B + 19*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*a*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(3/2)) + ((A - B + 2*C)*sin(c + d*x))/(2*a*d*cos(c + d*x)^(5/2)*sqrt(a + a*sec(c + d*x))) - ((2*A - 6*B + 7*C)*sin(c + d*x))/(4*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))],


[sec(c + d*x)^4*(a + a*sec(c + d*x)), x, 5, (3*a*arctanh(sin(c + d*x)))/(8*d) + (2*a*tan(c + d*x))/(3*d) + (3*a*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^3*(a + a*sec(c + d*x)), x, 4, (a*arctanh(sin(c + d*x)))/(2*d) + (2*a*tan(c + d*x))/(3*d) + (a*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + a*sec(c + d*x)), x, 3, (a*arctanh(sin(c + d*x)))/(2*d) + (a*tan(c + d*x))/d + (a*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^1*(a + a*sec(c + d*x)), x, 2, (a*arctanh(sin(c + d*x)))/d + (a*tan(c + d*x))/d],
[sec(c + d*x)^0*(a + a*sec(c + d*x)), x, 2, a*x + (a*arctanh(sin(c + d*x)))/d],
[cos(c + d*x)^1*(a + a*sec(c + d*x)), x, 2, a*x + (a*sin(c + d*x))/d],
[cos(c + d*x)^2*(a + a*sec(c + d*x)), x, 3, (a*x)/2 + (a*sin(c + d*x))/d + (a*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + a*sec(c + d*x)), x, 4, (a*x)/2 + (2*a*sin(c + d*x))/(3*d) + (a*cos(c + d*x)*sin(c + d*x))/(2*d) + (a*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + a*sec(c + d*x)), x, 5, (3*a*x)/8 + (2*a*sin(c + d*x))/(3*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)],

[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 7, -((283*A - 163*B + 75*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((21*A - 13*B + 5*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) + ((2671*A - 1495*B + 735*C)*sin(c + d*x))/(240*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((787*A - 475*B + 195*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*sec(c + d*x))) + ((157*A - 85*B + 45*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 6, ((163*A - 75*B + 19*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*sec(c + d*x))^(5/2)) - ((17*A - 9*B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*sec(c + d*x))^(3/2)) - ((299*A - 147*B + 27*C)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + ((95*A - 39*B + 15*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*sec(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + a*sec(c + d*x))^(5/2), x, 5, -((75*A - 19*B - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(5/2)) - ((13*A - 5*B - 3*C)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*sec(c + d*x))^(3/2)) + ((49*A - 9*B + C)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(1/2)*(a + a*sec(c + d*x))^(5/2)), x, 4, ((19*A + 5*B + 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)) - ((9*A - B - 7*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(5/2)), x, 7, ((5*A + 3*B - 43*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (2*sqrt(2)*C*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)) + ((5*A + 3*B - 11*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*sec(c + d*x))^(3/2))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(5/2)), x, 8, ((3*A - 43*B + 115*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) + (sqrt(2)*(2*B - 5*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*sec(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(7/2)*(a + a*sec(c + d*x))^(5/2)) + ((A + 7*B - 15*C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(5/2)*(a + a*sec(c + d*x))^(3/2)) + ((3*A - 11*B + 35*C)*sin(c + d*x))/(16*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*sec(c + d*x)))]
]:
