\(\int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [328]

Optimal result

Integrand size = 31, antiderivative size = 407 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{2 b^5 d}+\frac {a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \] Output:

-1/2*(6*A*a*b-12*B*a^2-B*b^2)*arctanh(sin(d*x+c))/b^5/d+a^2*(6*A*a^4*b-15* 
A*a^2*b^3+12*A*b^5-12*B*a^5+29*B*a^3*b^2-20*B*a*b^4)*arctanh((a-b)^(1/2)*t 
an(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^5/(a+b)^(5/2)/d+1/2*(6*A*a^4* 
b-11*A*a^2*b^3+2*A*b^5-12*B*a^5+21*B*a^3*b^2-6*B*a*b^4)*tan(d*x+c)/b^4/(a^ 
2-b^2)^2/d-1/2*(3*A*a^3*b-6*A*a*b^3-6*B*a^4+10*B*a^2*b^2-B*b^4)*sec(d*x+c) 
*tan(d*x+c)/b^3/(a^2-b^2)^2/d+1/2*a*(A*b-B*a)*sec(d*x+c)^3*tan(d*x+c)/b/(a 
^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*a*(2*A*a^2*b-5*A*b^3-4*B*a^3+7*B*a*b^2)*s 
ec(d*x+c)^2*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))
 

Mathematica [A] (verified)

Time = 2.78 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {16 a^2 \left (-6 a^4 A b+15 a^2 A b^3-12 A b^5+12 a^5 B-29 a^3 b^2 B+20 a b^4 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-8 \left (-6 a A b+12 a^2 B+b^2 B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 \left (-6 a A b+12 a^2 B+b^2 B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 b \left (18 a^5 A b^2-32 a^3 A b^4+8 a A b^6-36 a^6 b B+68 a^4 b^3 B-30 a^2 b^5 B+4 b^7 B+\left (18 a^6 A b-25 a^4 A b^3-10 a^2 A b^5+8 A b^7-36 a^7 B+47 a^5 b^2 B+14 a^3 b^4 B-16 a b^6 B\right ) \cos (c+d x)-2 a b \left (-9 a^4 A b+16 a^2 A b^3-4 A b^5+18 a^5 B-32 a^3 b^2 B+11 a b^4 B\right ) \cos (2 (c+d x))+6 a^6 A b \cos (3 (c+d x))-11 a^4 A b^3 \cos (3 (c+d x))+2 a^2 A b^5 \cos (3 (c+d x))-12 a^7 B \cos (3 (c+d x))+21 a^5 b^2 B \cos (3 (c+d x))-6 a^3 b^4 B \cos (3 (c+d x))\right ) \sec (c+d x) \tan (c+d x)}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{16 b^5 d} \] Input:

Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]
 

Output:

((16*a^2*(-6*a^4*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 
 20*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - 
b^2)^(5/2) - 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c 
 + d*x)/2]] + 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[( 
c + d*x)/2]] + (2*b*(18*a^5*A*b^2 - 32*a^3*A*b^4 + 8*a*A*b^6 - 36*a^6*b*B 
+ 68*a^4*b^3*B - 30*a^2*b^5*B + 4*b^7*B + (18*a^6*A*b - 25*a^4*A*b^3 - 10* 
a^2*A*b^5 + 8*A*b^7 - 36*a^7*B + 47*a^5*b^2*B + 14*a^3*b^4*B - 16*a*b^6*B) 
*Cos[c + d*x] - 2*a*b*(-9*a^4*A*b + 16*a^2*A*b^3 - 4*A*b^5 + 18*a^5*B - 32 
*a^3*b^2*B + 11*a*b^4*B)*Cos[2*(c + d*x)] + 6*a^6*A*b*Cos[3*(c + d*x)] - 1 
1*a^4*A*b^3*Cos[3*(c + d*x)] + 2*a^2*A*b^5*Cos[3*(c + d*x)] - 12*a^7*B*Cos 
[3*(c + d*x)] + 21*a^5*b^2*B*Cos[3*(c + d*x)] - 6*a^3*b^4*B*Cos[3*(c + d*x 
)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(16 
*b^5*d)
 

Rubi [A] (verified)

Time = 2.91 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.06, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4517, 3042, 4586, 25, 3042, 4580, 27, 3042, 4570, 3042, 4486, 3042, 4257, 4318, 3042, 3138, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^5 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 4517

\(\displaystyle \frac {\int \frac {\sec ^3(c+d x) \left (-2 \left (-2 B a^2+A b a+b^2 B\right ) \sec ^2(c+d x)-2 b (A b-a B) \sec (c+d x)+3 a (A b-a B)\right )}{(a+b \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (-2 \left (-2 B a^2+A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (A b-a B)\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4586

\(\displaystyle \frac {\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\sec ^2(c+d x) \left (-2 \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right ) \sec ^2(c+d x)+b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)+2 a \left (-4 B a^3+2 A b a^2+7 b^2 B a-5 A b^3\right )\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\sec ^2(c+d x) \left (-2 \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right ) \sec ^2(c+d x)+b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)+2 a \left (-4 B a^3+2 A b a^2+7 b^2 B a-5 A b^3\right )\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (-2 \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-4 B a^3+2 A b a^2+7 b^2 B a-5 A b^3\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4580

\(\displaystyle \frac {\frac {\frac {\int -\frac {2 \sec (c+d x) \left (-\left (\left (-12 B a^5+6 A b a^4+21 b^2 B a^3-11 A b^3 a^2-6 b^4 B a+2 A b^5\right ) \sec ^2(c+d x)\right )-b \left (-2 B a^4+A b a^3+4 b^2 B a^2-4 A b^3 a+b^4 B\right ) \sec (c+d x)+a \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right )\right )}{a+b \sec (c+d x)}dx}{2 b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {\sec (c+d x) \left (-\left (\left (-12 B a^5+6 A b a^4+21 b^2 B a^3-11 A b^3 a^2-6 b^4 B a+2 A b^5\right ) \sec ^2(c+d x)\right )-b \left (-2 B a^4+A b a^3+4 b^2 B a^2-4 A b^3 a+b^4 B\right ) \sec (c+d x)+a \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right )\right )}{a+b \sec (c+d x)}dx}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\left (12 B a^5-6 A b a^4-21 b^2 B a^3+11 A b^3 a^2+6 b^4 B a-2 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-2 B a^4+A b a^3+4 b^2 B a^2-4 A b^3 a+b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4570

\(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\sec (c+d x) \left (\left (-12 B a^2+6 A b a-b^2 B\right ) \sec (c+d x) \left (a^2-b^2\right )^2+a b \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right )\right )}{a+b \sec (c+d x)}dx}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\left (-12 B a^2+6 A b a-b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+a b \left (-6 B a^4+3 A b a^3+10 b^2 B a^2-6 A b^3 a-b^4 B\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4486

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \int \sec (c+d x)dx}{b}-\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{b}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b}-\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b^2}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b^2}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {2 a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b^2 d}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{b d}-\frac {\frac {\frac {\left (a^2-b^2\right )^2 \left (-12 a^2 B+6 a A b-b^2 B\right ) \text {arctanh}(\sin (c+d x))}{b d}-\frac {2 a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\)

Input:

Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]
 

Output:

(a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[ 
c + d*x])^2) + ((a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x 
]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) + (-(((3*a^3*A*b 
- 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/( 
b*d)) - ((((a^2 - b^2)^2*(6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x] 
])/(b*d) - (2*a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3 
*b^2*B - 20*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/ 
(Sqrt[a - b]*b*Sqrt[a + b]*d))/b - ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 
12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(b*d))/b)/(b*(a^2 - b^2 
)))/(2*b*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4486

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[( 
e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[B/b   Int[Csc[e + f*x], 
 x], x] + Simp[(A*b - a*B)/b   Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x 
] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
 

rule 4517

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*d^2*( 
A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 
 2)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[d/(b*(m + 1)*(a^2 - b^2))   Int[( 
a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*( 
n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2 
*(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f 
, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[ 
n, 1]
 

rule 4570

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e 
_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S 
ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) 
)), x] + Simp[1/(b*(m + 2))   Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ 
b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; 
 FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]
 

rule 4580

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ 
(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x 
_Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 
1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[Csc[e + f*x]*(a + b*Csc[e 
+ f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* 
(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & 
& NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

rule 4586

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + 
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) 
), x] + Simp[d/(b*(a^2 - b^2)*(m + 1))   Int[(a + b*Csc[e + f*x])^(m + 1)*( 
d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A 
 - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n 
+ b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C 
}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 A b -6 B a -B b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-6 A a b +12 B \,a^{2}+b^{2} B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}-\frac {2 a^{2} \left (\frac {\frac {\left (4 A \,a^{2} b -A a \,b^{2}-8 A \,b^{3}-6 B \,a^{3}+B \,a^{2} b +10 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (4 A \,a^{2} b +A a \,b^{2}-8 A \,b^{3}-6 B \,a^{3}-B \,a^{2} b +10 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 A \,a^{4} b -15 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+29 B \,a^{3} b^{2}-20 B a \,b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 A b -6 B a -B b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (6 A a b -12 B \,a^{2}-b^{2} B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}}{d}\)	\(464\)
default	\(\frac {-\frac {B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 A b -6 B a -B b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-6 A a b +12 B \,a^{2}+b^{2} B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{5}}-\frac {2 a^{2} \left (\frac {\frac {\left (4 A \,a^{2} b -A a \,b^{2}-8 A \,b^{3}-6 B \,a^{3}+B \,a^{2} b +10 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (4 A \,a^{2} b +A a \,b^{2}-8 A \,b^{3}-6 B \,a^{3}-B \,a^{2} b +10 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 A \,a^{4} b -15 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+29 B \,a^{3} b^{2}-20 B a \,b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}+\frac {B}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 A b -6 B a -B b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (6 A a b -12 B \,a^{2}-b^{2} B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{5}}}{d}\)	\(464\)
risch	\(\text {Expression too large to display}\)	\(2118\)

Input:

int(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBO 
SE)
 

Output:

1/d*(-1/2*B/b^3/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(2*A*b-6*B*a-B*b)/b^4/(tan(1/ 
2*d*x+1/2*c)+1)+1/2/b^5*(-6*A*a*b+12*B*a^2+B*b^2)*ln(tan(1/2*d*x+1/2*c)+1) 
-2*a^2/b^5*((1/2*(4*A*a^2*b-A*a*b^2-8*A*b^3-6*B*a^3+B*a^2*b+10*B*a*b^2)*a* 
b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*a*(4*A*a^2*b+A*a*b^2-8* 
A*b^3-6*B*a^3-B*a^2*b+10*B*a*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(tan(1 
/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-1/2*(6*A*a^4*b-15*A*a^2*b^ 
3+12*A*b^5-12*B*a^5+29*B*a^3*b^2-20*B*a*b^4)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a 
-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+1/2*B/b^ 
3/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(2*A*b-6*B*a-B*b)/b^4/(tan(1/2*d*x+1/2*c)-1 
)+1/2*(6*A*a*b-12*B*a^2-B*b^2)/b^5*ln(tan(1/2*d*x+1/2*c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1194 vs. \(2 (387) = 774\).

Time = 38.77 (sec) , antiderivative size = 2444, normalized size of antiderivative = 6.00 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="f 
ricas")
 

Output:

[-1/4*(((12*B*a^9 - 6*A*a^8*b - 29*B*a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 
 - 12*A*a^4*b^5)*cos(d*x + c)^4 + 2*(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*b 
^3 + 15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c)^3 + (12*B*a^ 
7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4 + 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12*A*a^ 
2*b^7)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2* 
b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) 
+ 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - ((12*B*a 
^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A*a^5*b^ 
5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*a^9*b 
- 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 
- 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^2 - 6* 
A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^7 - 9* 
B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + (( 
12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A* 
a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B* 
a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^ 
4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^ 
2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^ 
7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 
1) - 2*(B*a^6*b^4 - 3*B*a^4*b^6 + 3*B*a^2*b^8 - B*b^10 - (12*B*a^9*b - ...
 

Sympy [F]

\[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(sec(d*x+c)**5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**3,x)
 

Output:

Integral((A + B*sec(c + d*x))*sec(c + d*x)**5/(a + b*sec(c + d*x))**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="m 
axima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1391 vs. \(2 (387) = 774\).

Time = 0.28 (sec) , antiderivative size = 1391, normalized size of antiderivative = 3.42 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="g 
iac")
 

Output:

-1/2*(2*(12*B*a^7 - 6*A*a^6*b - 29*B*a^5*b^2 + 15*A*a^4*b^3 + 20*B*a^3*b^4 
 - 12*A*a^2*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arcta 
n(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(( 
a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(-a^2 + b^2)) - 2*(12*B*a^7*tan(1/2*d*x + 1 
/2*c)^7 - 6*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 18*B*a^6*b*tan(1/2*d*x + 1/2* 
c)^7 + 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 17*B*a^5*b^2*tan(1/2*d*x + 1/2 
*c)^7 + 9*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*B*a^4*b^3*tan(1/2*d*x + 1/ 
2*c)^7 - 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 2*B*a^3*b^4*tan(1/2*d*x + 1 
/2*c)^7 + 2*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 13*B*a^2*b^5*tan(1/2*d*x + 
1/2*c)^7 + 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 4*B*a*b^6*tan(1/2*d*x + 1/2* 
c)^7 - 2*A*b^7*tan(1/2*d*x + 1/2*c)^7 + B*b^7*tan(1/2*d*x + 1/2*c)^7 - 36* 
B*a^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 18*B*a^ 
6*b*tan(1/2*d*x + 1/2*c)^5 - 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 67*B*a^5 
*b^2*tan(1/2*d*x + 1/2*c)^5 - 35*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*B*a 
^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 26*B 
*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 10*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 5* 
B*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 4*B* 
a*b^6*tan(1/2*d*x + 1/2*c)^5 - 2*A*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*B*b^7*ta 
n(1/2*d*x + 1/2*c)^5 + 36*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 18*A*a^6*b*tan(1/ 
2*d*x + 1/2*c)^3 + 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 9*A*a^5*b^2*tan(...
 

Mupad [B] (verification not implemented)

Time = 23.21 (sec) , antiderivative size = 10533, normalized size of antiderivative = 25.88 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:

int((A + B/cos(c + d*x))/(cos(c + d*x)^5*(a + b/cos(c + d*x))^3),x)
 

Output:

((tan(c/2 + (d*x)/2)^3*(2*A*b^7 + 36*B*a^7 + 3*B*b^7 - 10*A*a^2*b^5 + 16*A 
*a^3*b^4 + 35*A*a^4*b^3 - 9*A*a^5*b^2 + 5*B*a^2*b^5 + 26*B*a^3*b^4 - 29*B* 
a^4*b^3 - 67*B*a^5*b^2 - 4*A*a*b^6 - 18*A*a^6*b - 4*B*a*b^6 + 18*B*a^6*b)) 
/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) + (tan(c/2 + (d*x)/2)^5*(3*B*b^7 - 
36*B*a^7 - 2*A*b^7 + 10*A*a^2*b^5 + 16*A*a^3*b^4 - 35*A*a^4*b^3 - 9*A*a^5* 
b^2 + 5*B*a^2*b^5 - 26*B*a^3*b^4 - 29*B*a^4*b^3 + 67*B*a^5*b^2 - 4*A*a*b^6 
 + 18*A*a^6*b + 4*B*a*b^6 + 18*B*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b 
^4)) - (tan(c/2 + (d*x)/2)^7*(B*b^6 - 12*B*a^6 - 2*A*b^6 + 4*A*a^2*b^4 - 1 
2*A*a^3*b^3 - 3*A*a^4*b^2 - 8*B*a^2*b^4 - 10*B*a^3*b^3 + 23*B*a^4*b^2 + 2* 
A*a*b^5 + 6*A*a^5*b + 5*B*a*b^5 + 6*B*a^5*b))/((a*b^4 - b^5)*(a + b)^2) + 
(tan(c/2 + (d*x)/2)*(2*A*b^6 - 12*B*a^6 + B*b^6 - 4*A*a^2*b^4 - 12*A*a^3*b 
^3 + 3*A*a^4*b^2 - 8*B*a^2*b^4 + 10*B*a^3*b^3 + 23*B*a^4*b^2 + 2*A*a*b^5 + 
 6*A*a^5*b - 5*B*a*b^5 - 6*B*a^5*b))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)))/ 
(d*(2*a*b + tan(c/2 + (d*x)/2)^4*(6*a^2 - 2*b^2) - tan(c/2 + (d*x)/2)^2*(4 
*a*b + 4*a^2) + tan(c/2 + (d*x)/2)^6*(4*a*b - 4*a^2) + tan(c/2 + (d*x)/2)^ 
8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((8*tan(c/2 + (d*x)/2)*(288* 
B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72 
*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432* 
A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^ 
2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1637, normalized size of antiderivative = 4.02 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x)
 

Output:

( - 12*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b) 
/sqrt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**6 + 16*sqrt( - a**2 
 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b** 
2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**2 + 12*sqrt( - a**2 + b**2)*atan( 
(tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d* 
x)*a**6 - 16*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x) 
/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a**4*b**2 - 12*sqrt( - a**2 + b* 
*2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*s 
in(c + d*x)**2*a**5*b + 16*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - 
 tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*sin(c + d*x)**2*a**3*b**3 + 12* 
sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( 
- a**2 + b**2))*a**5*b - 16*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a 
- tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a**3*b**3 - 6*cos(c + d*x)*log 
(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**7 + 11*cos(c + d*x)*log(tan((c + 
 d*x)/2) - 1)*sin(c + d*x)**2*a**5*b**2 - 4*cos(c + d*x)*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**2*a**3*b**4 - cos(c + d*x)*log(tan((c + d*x)/2) - 1 
)*sin(c + d*x)**2*a*b**6 + 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**7 - 
 11*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**5*b**2 + 4*cos(c + d*x)*log( 
tan((c + d*x)/2) - 1)*a**3*b**4 + cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a 
*b**6 + 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**7 -...