\(\int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [281]

Optimal result

Integrand size = 28, antiderivative size = 698 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{4 a d^4}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \] Output:

5/2*I*f^3*polylog(2,-I*exp(I*(d*x+c)))/a/d^4-1/2*I*f^3*polylog(2,-exp(2*I* 
(d*x+c)))/a/d^4-1/2*I*f*(f*x+e)^2/a/d^2+f^2*(f*x+e)*ln(1+exp(2*I*(d*x+c))) 
/a/d^3-3/4*I*(f*x+e)^3*arctan(exp(I*(d*x+c)))/a/d-9/8*I*f*(f*x+e)^2*polylo 
g(2,I*exp(I*(d*x+c)))/a/d^2+9/4*I*f^3*polylog(4,I*exp(I*(d*x+c)))/a/d^4-9/ 
4*I*f^3*polylog(4,-I*exp(I*(d*x+c)))/a/d^4-5*I*f^2*(f*x+e)*arctan(exp(I*(d 
*x+c)))/a/d^3-9/4*f^2*(f*x+e)*polylog(3,-I*exp(I*(d*x+c)))/a/d^3+9/4*f^2*( 
f*x+e)*polylog(3,I*exp(I*(d*x+c)))/a/d^3-5/2*I*f^3*polylog(2,I*exp(I*(d*x+ 
c)))/a/d^4+9/8*I*f*(f*x+e)^2*polylog(2,-I*exp(I*(d*x+c)))/a/d^2-1/4*f^3*se 
c(d*x+c)/a/d^4-9/8*f*(f*x+e)^2*sec(d*x+c)/a/d^2-1/4*f^2*(f*x+e)*sec(d*x+c) 
^2/a/d^3-1/4*f*(f*x+e)^2*sec(d*x+c)^3/a/d^2-1/4*(f*x+e)^3*sec(d*x+c)^4/a/d 
+1/4*f^3*tan(d*x+c)/a/d^4+1/2*f*(f*x+e)^2*tan(d*x+c)/a/d^2+1/4*f^2*(f*x+e) 
*sec(d*x+c)*tan(d*x+c)/a/d^3+3/8*(f*x+e)^3*sec(d*x+c)*tan(d*x+c)/a/d+1/4*f 
*(f*x+e)^2*sec(d*x+c)^2*tan(d*x+c)/a/d^2+1/4*(f*x+e)^3*sec(d*x+c)^3*tan(d* 
x+c)/a/d
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2278\) vs. \(2(698)=1396\).

Time = 12.73 (sec) , antiderivative size = 2278, normalized size of antiderivative = 3.26 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Result too large to show} \] Input:

Integrate[((e + f*x)^3*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
 

Output:

(-3*(6*d^4*e^2*f*x^2 + 8*d^2*f^3*x^2 + 4*d^4*e*f^2*x^3 + d^4*f^3*x^4 - (4* 
I)*d^4*e^3*x*Cos[c] - (16*I)*d^2*e*f^2*x*Cos[c] - (4*I)*d^3*e^3*Log[-Cos[c 
 + d*x] - I*(-1 + Sin[c + d*x])] - (16*I)*d*e*f^2*Log[-Cos[c + d*x] - I*(- 
1 + Sin[c + d*x])] - (12*I)*d^3*e^2*f*x*Log[1 - I*Cos[c + d*x] - Sin[c + d 
*x]] - (16*I)*d*f^3*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (12*I)*d^3* 
e*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (4*I)*d^3*f^3*x^3*Log[1 
 - I*Cos[c + d*x] - Sin[c + d*x]] - 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin 
[c + d*x]] + 24*d*f^2*(e + f*x)*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]]* 
(Cos[c] + I*(-1 + Sin[c])) + 4*f*(4*f^2 + 3*d^2*(e + f*x)^2)*PolyLog[2, I* 
Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] - Sin[c]) + 4*d^3*e^3*Log[-Cos[ 
c + d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] + I*Sin[c]) + 16*d*e*f^2*Log[-Co 
s[c + d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] + I*Sin[c]) + 12*d^3*e^2*f*x*L 
og[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 16*d*f^3*x*Log 
[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 12*d^3*e*f^2*x^2 
*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*d^3*f^3*x^ 
3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*d^4*e^3*x 
*Sin[c] + 16*d^2*e*f^2*x*Sin[c] + 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin[c 
 + d*x]]*((-I)*Cos[c] + Sin[c])))/(32*a*d^4*(Cos[c] + I*(-1 + Sin[c]))) - 
((Cos[c] + I*Sin[c])*(((28*f^2 + 3*d^2*(e + f*x)^2)^2*(Cos[c] - I*Sin[c])) 
/(12*d^2*f) + (f*(9*d^2*e^2 + 28*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Si...
 

Rubi [F]

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.

\(\displaystyle \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 5042

\(\displaystyle \frac {\int (e+f x)^3 \sec ^5(c+d x)dx}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\frac {f^2 \int (e+f x) \sec ^3(c+d x)dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \sec ^3(c+d x)dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {f^2 \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {\frac {f^2 \left (\frac {1}{2} \int (e+f x) \sec (c+d x)dx-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {f^2 \left (\frac {1}{2} \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {f^2 \left (\frac {1}{2} \left (-\frac {f \int \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {f^2 \left (\frac {1}{2} \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \int (e+f x) \sec (c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \sec (c+d x)dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \int (e+f x) \csc \left (c+d x+\frac {\pi }{2}\right )dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \left (-\frac {f \int \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {3 f^2 \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3 f \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right )dx}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x)dx}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4909

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \sec ^4(c+d x)dx}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^4dx}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \sec ^2(c+d x)dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \sec ^2(c+d x)dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (-\frac {f^2 \int 1d(-\tan (c+d x))}{3 d^3}+\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \int (e+f x)^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {2 f \int -((e+f x) \tan (c+d x))dx}{d}+\frac {(e+f x)^2 \tan (c+d x)}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \int (e+f x) \tan (c+d x)dx}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}+\frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}}dx\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{2 i (c+d x)}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-2 i (c+d x)} \log \left (1+e^{2 i (c+d x)}\right )de^{2 i (c+d x)}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )+\frac {f^2 \tan (c+d x)}{3 d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \tan (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{d}\right )+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}-\frac {3 f (e+f x)^2 \sec (c+d x)}{2 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \tan (c+d x) \sec (c+d x)}{2 d}\right )}{2 d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {f^2 \tan (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}\right )\right )}{d}\right )-\frac {f (e+f x) \sec ^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )}{4 d}}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {f^2 \left (\frac {1}{2} \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )-\frac {f \sec (c+d x)}{2 d^2}+\frac {(e+f x) \sec (c+d x) \tan (c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \left (\frac {\sec (c+d x) \tan (c+d x) (e+f x)^3}{2 d}-\frac {3 f \sec (c+d x) (e+f x)^2}{2 d^2}+\frac {3 f^2 \left (-\frac {2 i (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (-\frac {2 i \arctan \left (e^{i (c+d x)}\right ) (e+f x)^3}{d}+\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}\right )\right )}{a}-\frac {\frac {(e+f x)^3 \sec ^4(c+d x)}{4 d}-\frac {3 f \left (\frac {\tan (c+d x) f^2}{3 d^3}-\frac {(e+f x) \sec ^2(c+d x) f}{3 d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {2}{3} \left (\frac {(e+f x)^2 \tan (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {i (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{4 d^2}\right )\right )}{d}\right )\right )}{4 d}}{a}\)

Input:

Int[((e + f*x)^3*Sec[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 4909

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b 
_.)*(x_)]^(p_.), x_Symbol] :> Simp[(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - 
 Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; FreeQ[{ 
a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 5042

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_. 
)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Sec[c + 
 d*x]^(n + 2), x], x] - Simp[1/b   Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*Tan 
[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a 
^2 - b^2, 0]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2040 vs. \(2 (612 ) = 1224\).

Time = 3.19 (sec) , antiderivative size = 2041, normalized size of antiderivative = 2.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(2041\)

Input:

int((f*x+e)^3*sec(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-9/4*I*f^3*polylog(4,-I*exp(I*(d*x+c)))/a/d^4-3/2/a/d^3*f^3*ln(1+I*exp(I*( 
d*x+c)))*x-3/2/a/d^4*f^3*ln(1+I*exp(I*(d*x+c)))*c+5*I/a/d^4*f^3*c*arctan(e 
xp(I*(d*x+c)))-5*I/a/d^3*e*f^2*arctan(exp(I*(d*x+c)))+9/8*I/a/d^2*e^2*f*po 
lylog(2,-I*exp(I*(d*x+c)))-2*I/a/d^3*f^3*c*x-9/8*I/a/d^2*f^3*polylog(2,I*e 
xp(I*(d*x+c)))*x^2+9/8*I/a/d^2*f^3*polylog(2,-I*exp(I*(d*x+c)))*x^2-9/8*I/ 
a/d^2*e^2*f*polylog(2,I*exp(I*(d*x+c)))+3/4*I/a/d^4*f^3*c^3*arctan(exp(I*( 
d*x+c)))-1/4*I*(2*f^3+2*d^3*f^3*x^3*exp(3*I*(d*x+c))+4*d^2*f^3*x^2+8*d^2*e 
*f^2*x+6*I*d^3*e^3*exp(4*I*(d*x+c))+9*d^3*e*f^2*x^2*exp(5*I*(d*x+c))+9*d^3 
*e^2*f*x*exp(5*I*(d*x+c))+44*d^2*e*f^2*x*exp(2*I*(d*x+c))+4*f^3*exp(2*I*(d 
*x+c))+18*d^2*e^2*f*exp(4*I*(d*x+c))+4*d*f^3*x*exp(3*I*(d*x+c))+4*d*e*f^2* 
exp(3*I*(d*x+c))+22*d^2*f^3*x^2*exp(2*I*(d*x+c))+22*d^2*e^2*f*exp(2*I*(d*x 
+c))-6*I*d^3*e^3*exp(2*I*(d*x+c))+2*d*f^3*x*exp(5*I*(d*x+c))+2*d*e*f^2*exp 
(5*I*(d*x+c))+3*d^3*f^3*x^3*exp(5*I*(d*x+c))+18*d^2*f^3*x^2*exp(4*I*(d*x+c 
))+2*d*f^3*x*exp(I*(d*x+c))+2*d*e*f^2*exp(I*(d*x+c))+3*d^3*f^3*x^3*exp(I*( 
d*x+c))+2*I*d^2*e*f^2*x*exp(I*(d*x+c))+2*f^3*exp(4*I*(d*x+c))+4*d^2*e^2*f- 
4*I*f^3*exp(3*I*(d*x+c))+3*d^3*e^3*exp(5*I*(d*x+c))+3*d^3*e^3*exp(I*(d*x+c 
))-2*I*f^3*exp(I*(d*x+c))-18*I*d^3*e*f^2*x^2*exp(2*I*(d*x+c))-18*I*d^3*e^2 
*f*x*exp(2*I*(d*x+c))-16*I*d^2*e*f^2*x*exp(3*I*(d*x+c))+6*d^3*e*f^2*x^2*ex 
p(3*I*(d*x+c))+6*d^3*e^2*f*x*exp(3*I*(d*x+c))-6*I*d^3*f^3*x^3*exp(2*I*(d*x 
+c))+I*d^2*f^3*x^2*exp(I*(d*x+c))+I*d^2*e^2*f*exp(I*(d*x+c))+9*d^3*e^2*...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2572 vs. \(2 (589) = 1178\).

Time = 0.25 (sec) , antiderivative size = 2572, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^3*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

Output:

1/16*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 6*d^3*e^2*f*x + 2*d^3*e^3 - 4*(2*d 
^2*f^3*x^2 + 4*d^2*e*f^2*x + 2*d^2*e^2*f + f^3)*cos(d*x + c)^3 - 2*(3*d^3* 
f^3*x^3 + 9*d^3*e*f^2*x^2 + 3*d^3*e^3 + 2*d*e*f^2 + (9*d^3*e^2*f + 2*d*f^3 
)*x)*cos(d*x + c)^2 - 14*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*cos(d*x 
 + c) - 3*((3*I*d^2*f^3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2*e^2*f + 4*I*f^3)*c 
os(d*x + c)^2*sin(d*x + c) + (3*I*d^2*f^3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2* 
e^2*f + 4*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) + sin(d*x + c)) + (( 
-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f - 28*I*f^3)*cos(d*x + 
c)^2*sin(d*x + c) + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f - 
 28*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) - sin(d*x + c)) - 3*((-3*I 
*d^2*f^3*x^2 - 6*I*d^2*e*f^2*x - 3*I*d^2*e^2*f - 4*I*f^3)*cos(d*x + c)^2*s 
in(d*x + c) + (-3*I*d^2*f^3*x^2 - 6*I*d^2*e*f^2*x - 3*I*d^2*e^2*f - 4*I*f^ 
3)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) + sin(d*x + c)) + ((9*I*d^2*f^3*x 
^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f + 28*I*f^3)*cos(d*x + c)^2*sin(d*x + 
 c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f + 28*I*f^3)*cos( 
d*x + c)^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) + ((3*d^3*e^3 - 9*c*d^2* 
e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)*f^3)*cos(d*x + c)^2*sin(d*x 
+ c) + (3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)* 
f^3)*cos(d*x + c)^2)*log(cos(d*x + c) + I*sin(d*x + c) + I) - 3*((d^3*e^3 
- 3*c*d^2*e^2*f + (3*c^2 + 4)*d*e*f^2 - (c^3 + 4*c)*f^3)*cos(d*x + c)^2...
 

Sympy [F]

\[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

integrate((f*x+e)**3*sec(d*x+c)**3/(a+a*sin(d*x+c)),x)
 

Output:

(Integral(e**3*sec(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**3*x**3 
*sec(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*sec(c + d 
*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sec(c + d*x)**3/(sin(c 
 + d*x) + 1), x))/a
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((f*x+e)^3*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sec \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^3*sec(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

integrate((f*x + e)^3*sec(d*x + c)^3/(a*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \] Input:

int((e + f*x)^3/(cos(c + d*x)^3*(a + a*sin(c + d*x))),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)^3*sec(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Output:

( - 87204*cos(c + d*x)*sin(c + d*x)**2*d**2*e**2*f - 174408*cos(c + d*x)*s 
in(c + d*x)**2*d**2*e*f**2*x - 87204*cos(c + d*x)*sin(c + d*x)**2*d**2*f** 
3*x**2 - 148824*cos(c + d*x)*sin(c + d*x)**2*d*e*f**2 + 1013896*cos(c + d* 
x)*sin(c + d*x)**2*d*f**3*x + 940012*cos(c + d*x)*sin(c + d*x)**2*f**3 + 3 
6504*cos(c + d*x)*sin(c + d*x)*d**3*e**2*f*x + 36504*cos(c + d*x)*sin(c + 
d*x)*d**3*e*f**2*x**2 + 12168*cos(c + d*x)*sin(c + d*x)*d**3*f**3*x**3 + 2 
2308*cos(c + d*x)*sin(c + d*x)*d**2*e**2*f + 112008*cos(c + d*x)*sin(c + d 
*x)*d**2*e*f**2*x + 56004*cos(c + d*x)*sin(c + d*x)*d**2*f**3*x**2 - 19656 
*cos(c + d*x)*sin(c + d*x)*d*e*f**2 + 182848*cos(c + d*x)*sin(c + d*x)*d*f 
**3*x + 40252*cos(c + d*x)*sin(c + d*x)*f**3 + 73008*cos(c + d*x)*d**3*e** 
2*f*x + 73008*cos(c + d*x)*d**3*e*f**2*x**2 + 24336*cos(c + d*x)*d**3*f**3 
*x**3 + 125736*cos(c + d*x)*d**2*e**2*f + 386256*cos(c + d*x)*d**2*e*f**2* 
x + 193128*cos(c + d*x)*d**2*f**3*x**2 + 134784*cos(c + d*x)*d*e*f**2 - 13 
20560*cos(c + d*x)*d*f**3*x - 936176*cos(c + d*x)*f**3 - 194688*int((tan(( 
c + d*x)/2)*x**3)/(tan((c + d*x)/2)**8 + 2*tan((c + d*x)/2)**7 - 2*tan((c 
+ d*x)/2)**6 - 6*tan((c + d*x)/2)**5 + 6*tan((c + d*x)/2)**3 + 2*tan((c + 
d*x)/2)**2 - 2*tan((c + d*x)/2) - 1),x)*sin(c + d*x)**3*d**4*f**3 - 194688 
*int((tan((c + d*x)/2)*x**3)/(tan((c + d*x)/2)**8 + 2*tan((c + d*x)/2)**7 
- 2*tan((c + d*x)/2)**6 - 6*tan((c + d*x)/2)**5 + 6*tan((c + d*x)/2)**3 + 
2*tan((c + d*x)/2)**2 - 2*tan((c + d*x)/2) - 1),x)*sin(c + d*x)**2*d**4...