\(\int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) [232]

Optimal result

Integrand size = 26, antiderivative size = 732 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^4} \] Output:

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

Mathematica [A] (verified)

Time = 3.99 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.22 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-2 d^3 (e+f x)^3 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))+\frac {b \left (3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )+i \left (2 i d^3 e^3 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+3 d^3 e^2 f x \log \left (1+\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )+3 d^3 e f^2 x^2 \log \left (1+\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )+d^3 f^3 x^3 \log \left (1+\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )-3 d^3 e^2 f x \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )-3 d^3 e f^2 x^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )-d^3 f^3 x^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )+3 i d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )+6 d f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )-6 d e f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )+6 i f^3 \operatorname {PolyLog}\left (4,-\frac {i b e^{i (c+d x)}}{-a+\sqrt {a^2-b^2}}\right )-6 i f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )\right )\right )}{\sqrt {a^2-b^2}}+3 i f \left (d^2 (e+f x)^2 \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))+2 i d f (e+f x) \operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))-2 f^2 \operatorname {PolyLog}(4,-\cos (c+d x)-i \sin (c+d x))\right )-3 i f \left (d^2 (e+f x)^2 \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))+2 i d f (e+f x) \operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))-2 f^2 \operatorname {PolyLog}(4,\cos (c+d x)+i \sin (c+d x))\right )}{a d^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.62 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5046, 3042, 3804, 2694, 27, 2620, 3011, 4671, 3011, 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.

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((-2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/d + (3*f*((I*(e + f*x)^2*PolyLo 
g[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 + S 
qrt[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]))/a
 

Defintions of rubi rules used

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 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 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 [F]

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

Input:

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

Output:

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

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. 3584 vs. \(2 (634) = 1268\).

Time = 0.40 (sec) , antiderivative size = 3584, normalized size of antiderivative = 4.90 \[ \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

( - 2*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*b 
*e**3 + int((csc(c + d*x)*x**3)/(sin(c + d*x)*b + a),x)*a**3*d*f**3 - int( 
(csc(c + d*x)*x**3)/(sin(c + d*x)*b + a),x)*a*b**2*d*f**3 + 3*int((csc(c + 
 d*x)*x**2)/(sin(c + d*x)*b + a),x)*a**3*d*e*f**2 - 3*int((csc(c + d*x)*x* 
*2)/(sin(c + d*x)*b + a),x)*a*b**2*d*e*f**2 + 3*int((csc(c + d*x)*x)/(sin( 
c + d*x)*b + a),x)*a**3*d*e**2*f - 3*int((csc(c + d*x)*x)/(sin(c + d*x)*b 
+ a),x)*a*b**2*d*e**2*f + log(tan((c + d*x)/2))*a**2*e**3 - log(tan((c + d 
*x)/2))*b**2*e**3)/(a*d*(a**2 - b**2))