\(\int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx\) [197]

Optimal result

Integrand size = 26, antiderivative size = 352 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \] Output:

I*(f*x+e)^3/a/d-2*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)^3*cot(1/2* 
c+1/4*Pi+1/2*d*x)/a/d-6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2+3*I*f*(f* 
x+e)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2+12*I*f^2*(f*x+e)*polylog(2,I*exp(I 
*(d*x+c)))/a/d^3-3*I*f*(f*x+e)^2*polylog(2,exp(I*(d*x+c)))/a/d^2-6*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^3-6*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
 

Mathematica [A] (warning: unable to verify)

Time = 4.36 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.26 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-2 (e+f x)^3 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))+\frac {3 i f \left (d^2 (e+f x)^2 \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))+2 i d f (e+f x) \operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))-2 f^2 \operatorname {PolyLog}(4,-\cos (c+d x)-i \sin (c+d x))\right )}{d^3}-\frac {3 i f \left (d^2 (e+f x)^2 \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))+2 i d f (e+f x) \operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))-2 f^2 \operatorname {PolyLog}(4,\cos (c+d x)+i \sin (c+d x))\right )}{d^3}+\frac {6 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{\cos (c)+i (1+\sin (c))}-\frac {2 (e+f x)^3 \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{a d} \] Input:

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

Output:

(-2*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + ((3*I)*f*(d^2*(e 
+ f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)* 
PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x 
] - I*Sin[c + d*x]]))/d^3 - ((3*I)*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c + d 
*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Si 
n[c + d*x]] - 2*f^2*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/d^3 + (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))/(Co 
s[c] + I*(1 + Sin[c])) - (2*(e + f*x)^3*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2 
])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(a*d)
 

Rubi [A] (verified)

Time = 2.16 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {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.

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3799

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {6 f \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {6 f \int -(e+f x)^2 \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {6 f \int (e+f x)^2 \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)^2}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \int (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{d}\right )}{d}\right )}{d}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}\)

Input:

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

Output:

-1/2*((-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)/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
 

Defintions of rubi rules used

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 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 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 3799

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])
 

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 4671

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]
 

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 5046

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]
 

rule 7143

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]
 

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. 1150 vs. \(2 (317 ) = 634\).

Time = 1.50 (sec) , antiderivative size = 1151, normalized size of antiderivative = 3.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(1151\)

Input:

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

Output:

6*I*f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e 
^3)/d/a/(exp(I*(d*x+c))+I)-6*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d^4+1/d/a* 
e^3*ln(exp(I*(d*x+c))-1)-1/d/a*e^3*ln(exp(I*(d*x+c))+1)+6/a/d^4*f^3*c^2*ln 
(exp(I*(d*x+c)))+6/a/d^2*f*e^2*ln(exp(I*(d*x+c)))-6/a/d^2*f^3*ln(1-I*exp(I 
*(d*x+c)))*x^2+6/a/d^4*f^3*c^2*ln(1-I*exp(I*(d*x+c)))+2*I/a/d*f^3*x^3-4*I/ 
a/d^4*f^3*c^3-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+12*I/a/d^2*f^2*e*c* 
x-12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c)))-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)) 
)*x+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))-6*I/a/d^3*f^3*c^2*x+12*I/ 
a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x+6*I/a/d*f^2*e*x^2+6*I/a/d^3*f^2*e* 
c^2-6*I/d^2/a*e*f^2*polylog(2,exp(I*(d*x+c)))*x+6*I/d^2/a*e*f^2*polylog(2, 
-exp(I*(d*x+c)))*x-12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+6/d^3/a*e*f^2*p 
olylog(3,exp(I*(d*x+c)))-6/d^3/a*e*f^2*polylog(3,-exp(I*(d*x+c)))-1/d^4/a* 
c^3*f^3*ln(exp(I*(d*x+c))-1)+1/d^4/a*c^3*f^3*ln(1-exp(I*(d*x+c)))-6/d^4/a* 
c^2*f^3*ln(exp(I*(d*x+c))+I)+1/d/a*f^3*ln(1-exp(I*(d*x+c)))*x^3+6/d^3/a*f^ 
3*polylog(3,exp(I*(d*x+c)))*x-1/d/a*f^3*ln(exp(I*(d*x+c))+1)*x^3-6/d^3/a*f 
^3*polylog(3,-exp(I*(d*x+c)))*x-6/d^2/a*e^2*f*ln(exp(I*(d*x+c))+I)+3/d/a*e 
^2*f*ln(1-exp(I*(d*x+c)))*x-3/d/a*e^2*f*ln(exp(I*(d*x+c))+1)*x+3/d^3/a*c^2 
*e*f^2*ln(exp(I*(d*x+c))-1)+3/d/a*e*f^2*ln(1-exp(I*(d*x+c)))*x^2-3/d/a*e*f 
^2*ln(exp(I*(d*x+c))+1)*x^2-3/d^3/a*e*f^2*ln(1-exp(I*(d*x+c)))*c^2+3/d^2/a 
*e^2*f*ln(1-exp(I*(d*x+c)))*c-3/d^2/a*c*e^2*f*ln(exp(I*(d*x+c))-1)+12/d...
 

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. 2924 vs. \(2 (305) = 610\).

Time = 0.19 (sec) , antiderivative size = 2924, normalized size of antiderivative = 8.31 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

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 + 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 + 2*I*d^2*e*f^2*x + I*d^2*e^2*f + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^ 
2*x + I*d^2*e^2*f)*cos(d*x + c) + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2 
*e^2*f)*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - 3*(-I*d^2*f^3 
*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - 
 I*d^2*e^2*f)*cos(d*x + c) + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2 
*f)*sin(d*x + c))*dilog(cos(d*x + c) - I*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) + (-I*d*f^3*x - I*d*e*f^ 
2)*sin(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) + (I*d*f^3*x + I*d*e*f^2)*s 
in(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 3*(I*d^2*f^3*x^2 + 2* 
I*d^2*e*f^2*x + I*d^2*e^2*f + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2*e^2 
*f)*cos(d*x + c) + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2*e^2*f)*sin(d*x 
 + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) - 3*(-I*d^2*f^3*x^2 - 2*I*d^2 
*e*f^2*x - I*d^2*e^2*f + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f)* 
cos(d*x + c) + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f)*sin(d*x + 
c))*dilog(-cos(d*x + c) - I*sin(d*x + c)) - (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 + (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) + (d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*...
 

Sympy [F]

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

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

Output:

(Integral(e**3*csc(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*cs 
c(c + d*x)/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*csc(c + d*x)/(s 
in(c + d*x) + 1), x) + Integral(3*e**2*f*x*csc(c + d*x)/(sin(c + d*x) + 1) 
, x))/a
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2796 vs. \(2 (305) = 610\).

Time = 0.51 (sec) , antiderivative size = 2796, normalized size of antiderivative = 7.94 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-(3*c*e^2*f*(2/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1)) + log(sin(d*x + 
 c)/(cos(d*x + c) + 1))/(a*d)) - e^3*(log(sin(d*x + c)/(cos(d*x + c) + 1)) 
/a + 2/(a + a*sin(d*x + c)/(cos(d*x + c) + 1))) + (12*I*c^2*d*e*f^2 - 4*I* 
c^3*f^3 - 12*(-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3 - (d^2*e^2*f - 2*c* 
d*e*f^2 + c^2*f^3)*cos(d*x + c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^ 
3)*sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 12*(I*(d*x + c) 
^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 
 - c*f^3)*(d*x + c))*cos(d*x + c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I* 
c*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) - 
2*(-3*I*c^2*d*e*f^2 - I*(d*x + c)^3*f^3 + I*c^3*f^3 + 3*(-I*d*e*f^2 + I*c* 
f^3)*(d*x + c)^2 + 3*(-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*(d*x + c) 
- (3*c^2*d*e*f^2 + (d*x + c)^3*f^3 - c^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + 
c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*x + c))*cos(d*x + c) + (-3 
*I*c^2*d*e*f^2 - I*(d*x + c)^3*f^3 + I*c^3*f^3 + 3*(-I*d*e*f^2 + I*c*f^3)* 
(d*x + c)^2 + 3*(-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*(d*x + c))*sin( 
d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) - 2*(3*I*c^2*d*e*f^2 - I 
*c^3*f^3 + (3*c^2*d*e*f^2 - c^3*f^3)*cos(d*x + c) + (3*I*c^2*d*e*f^2 - I*c 
^3*f^3)*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) - 1) - 2*(-I*(d*x 
 + c)^3*f^3 + 3*(-I*d*e*f^2 + I*c*f^3)*(d*x + c)^2 + 3*(-I*d^2*e^2*f + 2*I 
*c*d*e*f^2 - I*c^2*f^3)*(d*x + c) - ((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

(int((csc(c + d*x)*x**3)/(sin(c + d*x) + 1),x)*tan((c + d*x)/2)*d*f**3 + i 
nt((csc(c + d*x)*x**3)/(sin(c + d*x) + 1),x)*d*f**3 + 3*int((csc(c + d*x)* 
x**2)/(sin(c + d*x) + 1),x)*tan((c + d*x)/2)*d*e*f**2 + 3*int((csc(c + d*x 
)*x**2)/(sin(c + d*x) + 1),x)*d*e*f**2 + 3*int((csc(c + d*x)*x)/(sin(c + d 
*x) + 1),x)*tan((c + d*x)/2)*d*e**2*f + 3*int((csc(c + d*x)*x)/(sin(c + d* 
x) + 1),x)*d*e**2*f + log(tan((c + d*x)/2))*tan((c + d*x)/2)*e**3 + log(ta 
n((c + d*x)/2))*e**3 - 2*tan((c + d*x)/2)*e**3)/(a*d*(tan((c + d*x)/2) + 1 
))