Integrand size = 28, antiderivative size = 463 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \]
-12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3+2*(f*x+e)^3*arctanh(ex p(I*(d*x+c)))/a/d-(f*x+e)^3*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d-(f*x+e)^3*cot(d* x+c)/a/d+6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2+3*f*(f*x+e)^2*ln(1-exp (2*I*(d*x+c)))/a/d^2-3*I*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2-6*I* f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+6*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d ^4-3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a/d^3+6*f^2*(f*x+e)*polylog (3,-exp(I*(d*x+c)))/a/d^3+12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4-6*f^2*( f*x+e)*polylog(3,exp(I*(d*x+c)))/a/d^3+3/2*f^3*polylog(3,exp(2*I*(d*x+c))) /a/d^4+3*I*f*(f*x+e)^2*polylog(2,exp(I*(d*x+c)))/a/d^2-2*I*(f*x+e)^3/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1052\) vs. \(2(463)=926\).
Time = 8.61 (sec) , antiderivative size = 1052, normalized size of antiderivative = 2.27 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i d^3 e^2 (d e-3 f) x-i d^3 e^2 (d e+3 f) x-\frac {2 i d^3 (e+f x)^3}{-1+e^{2 i c}}-3 d^2 e (d e-2 f) f x \log \left (1-e^{-i (c+d x)}\right )-3 d^2 (d e-f) f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )-d^3 f^3 x^3 \log \left (1-e^{-i (c+d x)}\right )+3 d^2 e f (d e+2 f) x \log \left (1+e^{-i (c+d x)}\right )+3 d^2 f^2 (d e+f) x^2 \log \left (1+e^{-i (c+d x)}\right )+d^3 f^3 x^3 \log \left (1+e^{-i (c+d x)}\right )-d^2 e^2 (d e-3 f) \log \left (1-e^{i (c+d x)}\right )+d^2 e^2 (d e+3 f) \log \left (1+e^{i (c+d x)}\right )+3 i d e f (d e+2 f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+6 i d f^2 (d e+f) x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+3 i d^2 f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-3 i d e (d e-2 f) f \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-6 i d (d e-f) f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-3 i d^2 f^3 x^2 \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+6 f^2 (d e+f) \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )-6 (d e-f) f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 i f^3 \operatorname {PolyLog}\left (4,-e^{-i (c+d x)}\right )+6 i f^3 \operatorname {PolyLog}\left (4,e^{-i (c+d x)}\right )}{a d^4}-\frac {6 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{a d (\cos (c)+i (1+\sin (c)))}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {2 \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
(I*d^3*e^2*(d*e - 3*f)*x - I*d^3*e^2*(d*e + 3*f)*x - ((2*I)*d^3*(e + f*x)^ 3)/(-1 + E^((2*I)*c)) - 3*d^2*e*(d*e - 2*f)*f*x*Log[1 - E^((-I)*(c + d*x)) ] - 3*d^2*(d*e - f)*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] - d^3*f^3*x^3*Log[ 1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(d*e + 2*f)*x*Log[1 + E^((-I)*(c + d*x ))] + 3*d^2*f^2*(d*e + f)*x^2*Log[1 + E^((-I)*(c + d*x))] + d^3*f^3*x^3*Lo g[1 + E^((-I)*(c + d*x))] - d^2*e^2*(d*e - 3*f)*Log[1 - E^(I*(c + d*x))] + d^2*e^2*(d*e + 3*f)*Log[1 + E^(I*(c + d*x))] + (3*I)*d*e*f*(d*e + 2*f)*Po lyLog[2, -E^((-I)*(c + d*x))] + (6*I)*d*f^2*(d*e + f)*x*PolyLog[2, -E^((-I )*(c + d*x))] + (3*I)*d^2*f^3*x^2*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)* d*e*(d*e - 2*f)*f*PolyLog[2, E^((-I)*(c + d*x))] - (6*I)*d*(d*e - f)*f^2*x *PolyLog[2, E^((-I)*(c + d*x))] - (3*I)*d^2*f^3*x^2*PolyLog[2, E^((-I)*(c + d*x))] + 6*f^2*(d*e + f)*PolyLog[3, -E^((-I)*(c + d*x))] + 6*d*f^3*x*Pol yLog[3, -E^((-I)*(c + d*x))] - 6*(d*e - f)*f^2*PolyLog[3, E^((-I)*(c + d*x ))] - 6*d*f^3*x*PolyLog[3, E^((-I)*(c + d*x))] - (6*I)*f^3*PolyLog[4, -E^( (-I)*(c + d*x))] + (6*I)*f^3*PolyLog[4, E^((-I)*(c + d*x))])/(a*d^4) - (6* f*(Cos[c] + I*Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x )^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2 *f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog [3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(a *d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c/2]*Csc[c/2 + (d*x)/2]*(e^3*Sin[(...
Time = 3.50 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.16, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5046, 3042, 4672, 3042, 25, 4202, 2620, 3011, 2720, 5046, 3042, 3799, 3042, 4671, 3011, 4672, 3042, 25, 4202, 2620, 3011, 2720, 7143, 7163, 2720, 7143}
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.
| \(\displaystyle \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {\int (e+f x)^3 \csc ^2(c+d x)dx}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \csc (c+d x)^2dx}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \cot (c+d x)dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \int -(e+f x)^2 \tan \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)^2}{1+e^{i (2 c+2 d x+\pi )}}dx\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \int (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\int \frac {(e+f x)^3 \csc (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\) |
| \(\Big \downarrow \) 5046 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx-\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx-\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\int (e+f x)^3 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}-\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}-\frac {\int (e+f x)^3 \csc (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-\frac {3 f \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {3 f \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\frac {6 f \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {\frac {6 f \int -(e+f x)^2 \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {6 f \int (e+f x)^2 \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)^2}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \int (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \int e^{-i (2 c+2 d x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (3,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (3,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (3,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (3,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}+\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\) |
(-(((e + f*x)^3*Cot[c + d*x])/d) - (3*f*(((I/3)*(e + f*x)^3)/f - (2*I)*((( -1/2*I)*(e + f*x)^2*Log[1 + E^(I*(2*c + Pi + 2*d*x))])/d + (I*f*(((I/2)*(e + f*x)*PolyLog[2, -E^(I*(2*c + Pi + 2*d*x))])/d - (f*PolyLog[3, -E^(I*(2* c + Pi + 2*d*x))])/(4*d^2)))/d)))/d)/a + ((-2*(e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/d - (6*f*(((I/3)*(e + f*x)^3)/f - (2*I)*(((-I)*(e + f*x)^2*Log[ 1 + E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d + ((2*I)*f*((I*(e + f*x)*PolyLog[2, -E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d - (f*PolyLog[3, -E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d^2))/d)))/d)/(2*a) - ((-2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x) )])/d + (3*f*((I*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/d - ((2*I)*f*(( (-I)*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/d + (f*PolyLog[4, -E^(I*(c + d*x))])/d^2))/d))/d - (3*f*((I*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/d - ((2*I)*f*(((-I)*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/d + (f*PolyLog[4, E^(I*(c + d*x))])/d^2))/d))/d)/a
3.3.3.3.1 Defintions of rubi rules used
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]
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]]]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
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]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
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]
Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(Csc[c + d*x]^(n - 1)/(a + b*S in[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ [n, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1773 vs. \(2 (419 ) = 838\).
Time = 0.60 (sec) , antiderivative size = 1774, normalized size of antiderivative = 3.83
12*I/a/d^3*e*f^2*c*arctan(exp(I*(d*x+c)))-24*I/a/d^2*f^2*e*c*x+6*I/a/d^2*f ^2*e*polylog(2,exp(I*(d*x+c)))*x-6*I/a/d^2*f^2*e*polylog(2,-exp(I*(d*x+c)) )*x-3/a/d*f^2*e*ln(1-exp(I*(d*x+c)))*x^2+6/a/d^3*f^2*e*ln(1-exp(I*(d*x+c)) )*c+12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+6/a/d^2*f^2*e*ln(1-exp(I*(d*x+ c)))*x+12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x+6/a/d^2*f^2*e*ln(exp(I*(d*x +c))+1)*x+3/a/d^2*c*e^2*f*ln(exp(I*(d*x+c))-1)-6*I/a/d^2*e^2*f*arctan(exp( I*(d*x+c)))+3*I/a/d^2*e^2*f*polylog(2,exp(I*(d*x+c)))-3*I/a/d^2*e^2*f*poly log(2,-exp(I*(d*x+c)))-6*I/a/d^3*f^3*polylog(2,exp(I*(d*x+c)))*x-6*I/a/d^3 *f^3*polylog(2,-exp(I*(d*x+c)))*x-12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c) ))*x-6*I/a/d^4*c^2*f^3*arctan(exp(I*(d*x+c)))-12*I/a/d^3*e*f^2*c^2+12*I/a/ d^3*c^2*f^3*x-3*I/a/d^2*f^3*polylog(2,-exp(I*(d*x+c)))*x^2+3*I/a/d^2*f^3*p olylog(2,exp(I*(d*x+c)))*x^2-6*I/a/d^3*f^2*e*polylog(2,exp(I*(d*x+c)))-12* I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))-6*I/a/d^3*f^2*e*polylog(2,-exp(I *(d*x+c)))-12*I/a/d*e*f^2*x^2+24/a/d^3*e*f^2*c*ln(exp(I*(d*x+c)))-3/a/d^2* e^2*f*ln(1-exp(I*(d*x+c)))*c+3/a/d*e^2*f*ln(exp(I*(d*x+c))+1)*x-3/a/d*e^2* f*ln(1-exp(I*(d*x+c)))*x+3/a/d^3*c^2*f^2*e*ln(1-exp(I*(d*x+c)))-3/a/d^3*c^ 2*f^2*e*ln(exp(I*(d*x+c))-1)-6/a/d^3*c*f^2*e*ln(exp(I*(d*x+c))-1)-6/a/d^3* c*f^2*e*ln(1+exp(2*I*(d*x+c)))+3/a/d*f^2*e*ln(exp(I*(d*x+c))+1)*x^2-2*(-2* f^3*x^3+I*exp(I*(d*x+c))*f^3*x^3-6*e*f^2*x^2+3*I*exp(I*(d*x+c))*e*f^2*x^2- 6*e^2*f*x+3*I*exp(I*(d*x+c))*e^2*f*x-2*e^3+I*exp(I*(d*x+c))*e^3+f^3*x^3...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4799 vs. \(2 (405) = 810\).
Time = 0.45 (sec) , antiderivative size = 4799, normalized size of antiderivative = 10.37 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(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*(d^3 *f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c)^2 - 2*( d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c) - 3* (-I*d^2*f^3*x^2 - I*d^2*e^2*f + 2*I*d*e*f^2 + (I*d^2*f^3*x^2 + I*d^2*e^2*f - 2*I*d*e*f^2 + 2*I*(d^2*e*f^2 - d*f^3)*x)*cos(d*x + c)^2 - 2*I*(d^2*e*f^ 2 - d*f^3)*x + (-I*d^2*f^3*x^2 - I*d^2*e^2*f + 2*I*d*e*f^2 - 2*I*(d^2*e*f^ 2 - d*f^3)*x + (-I*d^2*f^3*x^2 - I*d^2*e^2*f + 2*I*d*e*f^2 - 2*I*(d^2*e*f^ 2 - d*f^3)*x)*cos(d*x + c))*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - 3*(I*d^2*f^3*x^2 + I*d^2*e^2*f - 2*I*d*e*f^2 + (-I*d^2*f^3*x^2 - I* d^2*e^2*f + 2*I*d*e*f^2 - 2*I*(d^2*e*f^2 - d*f^3)*x)*cos(d*x + c)^2 + 2*I* (d^2*e*f^2 - d*f^3)*x + (I*d^2*f^3*x^2 + I*d^2*e^2*f - 2*I*d*e*f^2 + 2*I*( d^2*e*f^2 - d*f^3)*x + (I*d^2*f^3*x^2 + I*d^2*e^2*f - 2*I*d*e*f^2 + 2*I*(d ^2*e*f^2 - d*f^3)*x)*cos(d*x + c))*sin(d*x + c))*dilog(cos(d*x + c) - I*si n(d*x + c)) - 12*(I*d*f^3*x + I*d*e*f^2 + (-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c)^2 + (I*d*f^3*x + I*d*e*f^2 + (I*d*f^3*x + I*d*e*f^2)*cos(d*x + c))*s in(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 12*(-I*d*f^3*x - I*d*e *f^2 + (I*d*f^3*x + I*d*e*f^2)*cos(d*x + c)^2 + (-I*d*f^3*x - I*d*e*f^2 + (-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c))*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 3*(-I*d^2*f^3*x^2 - I*d^2*e^2*f - 2*I*d*e*f^2 + (I*d^2* f^3*x^2 + I*d^2*e^2*f + 2*I*d*e*f^2 + 2*I*(d^2*e*f^2 + d*f^3)*x)*cos(d*...
\[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
(Integral(e**3*csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f**3*x**3 *csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*csc(c + d *x)**2/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*csc(c + d*x)**2/(sin(c + d*x) + 1), x))/a
Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \csc \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Hanged} \]