\(\int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [338]

Optimal result

Integrand size = 34, antiderivative size = 616 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^3}{3 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {2 i \left (a^2-b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1752\) vs. \(2(616)=1232\).

Time = 12.27 (sec) , antiderivative size = 1752, normalized size of antiderivative = 2.84 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5054

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 4910

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5054

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4904

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 5036

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

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

rule 2620

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

rule 2715

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

rule 2720

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

rule 2838

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

rule 3011

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

rule 3042

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

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

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

rule 3791

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

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 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 4908

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ 
(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr 
eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

rule 4910

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

rule 5036

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

rule 5054

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

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*cos(d*x+c)*cot(d*x+c)^2/(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. 2529 vs. \(2 (549) = 1098\).

Time = 0.32 (sec) , antiderivative size = 2529, normalized size of antiderivative = 4.11 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*a*b*d^2*f^2*x^2 + 4*a*b*d^2*e*f*x + 2*a*b*d^2*e^2 + 2*b^2*f^2*poly 
log(3, cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 2*b^2*f^2*polylog(3, 
cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*b^2*f^2*polylog(3, -cos(d* 
x + c) + I*sin(d*x + c))*sin(d*x + c) + 2*b^2*f^2*polylog(3, -cos(d*x + c) 
 - I*sin(d*x + c))*sin(d*x + c) + 2*(a^2 - b^2)*f^2*polylog(3, -(I*a*cos(d 
*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 
- b^2)/b^2))/b)*sin(d*x + c) + 2*(a^2 - b^2)*f^2*polylog(3, -(I*a*cos(d*x 
+ c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b 
^2)/b^2))/b)*sin(d*x + c) + 2*(a^2 - b^2)*f^2*polylog(3, -(-I*a*cos(d*x + 
c) + a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2 
)/b^2))/b)*sin(d*x + c) + 2*(a^2 - b^2)*f^2*polylog(3, -(-I*a*cos(d*x + c) 
 + a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/ 
b^2))/b)*sin(d*x + c) + 2*(-I*(a^2 - b^2)*d*f^2*x - I*(a^2 - b^2)*d*e*f)*d 
ilog((I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + 
c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 2*(-I*(a^2 - b^2)*d* 
f^2*x - I*(a^2 - b^2)*d*e*f)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) - (b 
*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d 
*x + c) + 2*(I*(a^2 - b^2)*d*f^2*x + I*(a^2 - b^2)*d*e*f)*dilog((-I*a*cos( 
d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 
 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 2*(I*(a^2 - b^2)*d*f^2*x + I*(a...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((f*x+e)^2*cos(d*x+c)*cot(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
 

Giac [F]

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

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

Output:

integrate((f*x + e)^2*cos(d*x + c)*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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