3.3.36 \(\int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [236]

3.3.36.1 Optimal result

Integrand size = 28, antiderivative size = 882 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a^2 d^4}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^4} \]

output

I*b^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d/(a^2-b^ 
2)^(1/2)+2*b*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a^2/d-(f*x+e)^3*cot(d*x+c)/ 
a/d+3*f*(f*x+e)^2*ln(1-exp(2*I*(d*x+c)))/a/d^2-I*b^2*(f*x+e)^3*ln(1-I*b*ex 
p(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d/(a^2-b^2)^(1/2)-3*I*f^2*(f*x+e)*po 
lylog(2,exp(2*I*(d*x+c)))/a/d^3-6*I*b^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d 
*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d^3/(a^2-b^2)^(1/2)+6*b*f^2*(f*x+e)*polylo 
g(3,-exp(I*(d*x+c)))/a^2/d^3-6*b*f^2*(f*x+e)*polylog(3,exp(I*(d*x+c)))/a^2 
/d^3+3/2*f^3*polylog(3,exp(2*I*(d*x+c)))/a/d^4-6*I*b*f^3*polylog(4,exp(I*( 
d*x+c)))/a^2/d^4+6*I*b*f^3*polylog(4,-exp(I*(d*x+c)))/a^2/d^4+3*I*b*f*(f*x 
+e)^2*polylog(2,exp(I*(d*x+c)))/a^2/d^2-I*(f*x+e)^3/a/d-3*b^2*f*(f*x+e)^2* 
polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/d^2/(a^2-b^2)^(1/2)+ 
3*b^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d^ 
2/(a^2-b^2)^(1/2)-3*I*b*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a^2/d^2+6*I 
*b^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/d^3 
/(a^2-b^2)^(1/2)+6*b^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2) 
))/a^2/d^4/(a^2-b^2)^(1/2)-6*b^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2- 
b^2)^(1/2)))/a^2/d^4/(a^2-b^2)^(1/2)
 

3.3.36.2 Mathematica [A] (warning: unable to verify)

Time = 9.07 (sec) , antiderivative size = 1735, normalized size of antiderivative = 1.97 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \]

input

Integrate[((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
 

output

(I*d^3*e^2*(b*d*e - 3*a*f)*x - I*d^3*e^2*(b*d*e + 3*a*f)*x - ((2*I)*a*d^3* 
(e + f*x)^3)/(-1 + E^((2*I)*c)) - 3*d^2*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^(( 
-I)*(c + d*x))] - 3*d^2*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))] 
- b*d^3*f^3*x^3*Log[1 - E^((-I)*(c + d*x))] + 3*d^2*e*f*(b*d*e + 2*a*f)*x* 
Log[1 + E^((-I)*(c + d*x))] + 3*d^2*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I)* 
(c + d*x))] + b*d^3*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] - d^2*e^2*(b*d*e - 
 3*a*f)*Log[1 - E^(I*(c + d*x))] + d^2*e^2*(b*d*e + 3*a*f)*Log[1 + E^(I*(c 
 + d*x))] + (3*I)*d*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I)*(c + d*x))] + 
(6*I)*d*f^2*(b*d*e + a*f)*x*PolyLog[2, -E^((-I)*(c + d*x))] + (3*I)*b*d^2* 
f^3*x^2*PolyLog[2, -E^((-I)*(c + d*x))] - (3*I)*d*e*f*(b*d*e - 2*a*f)*Poly 
Log[2, E^((-I)*(c + d*x))] - (6*I)*d*f^2*(b*d*e - a*f)*x*PolyLog[2, E^((-I 
)*(c + d*x))] - (3*I)*b*d^2*f^3*x^2*PolyLog[2, E^((-I)*(c + d*x))] + 6*f^2 
*(b*d*e + a*f)*PolyLog[3, -E^((-I)*(c + d*x))] + 6*b*d*f^3*x*PolyLog[3, -E 
^((-I)*(c + d*x))] + 6*f^2*(-(b*d*e) + a*f)*PolyLog[3, E^((-I)*(c + d*x))] 
 - 6*b*d*f^3*x*PolyLog[3, E^((-I)*(c + d*x))] - (6*I)*b*f^3*PolyLog[4, -E^ 
((-I)*(c + d*x))] + (6*I)*b*f^3*PolyLog[4, E^((-I)*(c + d*x))])/(a^2*d^4) 
+ (b^2*(2*Sqrt[-a^2 + b^2]*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a 
^2 - b^2]] + 3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((- 
I)*a + Sqrt[-a^2 + b^2])] + 3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b*E^( 
I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + Sqrt[a^2 - b^2]*d^3*f^3*x^...
 

3.3.36.3 Rubi [A] (verified)

Time = 4.29 (sec) , antiderivative size = 836, normalized size of antiderivative = 0.95, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5046, 3042, 4672, 3042, 25, 4202, 2620, 3011, 2720, 5046, 3042, 3804, 2694, 27, 2620, 3011, 4671, 3011, 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.

\(\displaystyle \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 5046

\(\displaystyle \frac {\int (e+f x)^3 \csc ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^3 \csc (c+d x)^2dx}{a}-\frac {b \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \cot (c+d x)dx}{d}-\frac {(e+f x)^3 \cot (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

\(\displaystyle -\frac {b \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^3 \cot (c+d x)}{d}-\frac {3 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)^2}{1+e^{i (2 c+2 d x+\pi )}}dx\right )}{d}}{a}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
 

output

(-(((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 - (b*(((-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 - (2*b*(((-1/2*I)*b*(((e + f*x 
)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) - (3*f*((I 
*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/d - 
((2*I)*f*(((-I)*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - 
 b^2])])/d + (f*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/d 
^2))/d))/(b*d)))/Sqrt[a^2 - b^2] + ((I/2)*b*(((e + f*x)^3*Log[1 - (I*b*E^( 
I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (3*f*((I*(e + f*x)^2*PolyLog 
[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d - ((2*I)*f*(((-I)*(e + 
 f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d + (f*Poly 
Log[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d^2))/d))/(b*d)))/Sqr 
t[a^2 - b^2]))/a))/a
 

3.3.36.3.1 Defintions of rubi rules used

rule 25

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

rule 27

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

rule 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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 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]
 

3.3.36.4 Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \left (\csc ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]

input

int((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

output

int((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

3.3.36.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]

input

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   Too 
many variables
 

3.3.36.6 Sympy [F]

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

input

integrate((f*x+e)**3*csc(d*x+c)**2/(a+b*sin(d*x+c)),x)
 

output

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

3.3.36.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

input

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima")
 

output

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

3.3.36.8 Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

input

integrate((f*x+e)^3*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")
 

output

Timed out
 

3.3.36.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

input

int((e + f*x)^3/(sin(c + d*x)^2*(a + b*sin(c + d*x))),x)
 

output

\text{Hanged}