\(\int \frac {(e+f x)^2 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) [326]

Optimal result

Integrand size = 32, antiderivative size = 557 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {(e+f x)^3}{3 b f}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {i \sqrt {a^2-b^2} (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b d}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {2 \sqrt {a^2-b^2} f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b d^2}+\frac {2 \sqrt {a^2-b^2} f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {2 i \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b d^3}+\frac {2 i \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b d^3} \] Output:

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

Mathematica [A] (verified)

Time = 2.39 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.09 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 b}+\frac {(e+f x)^2 \log \left (1-e^{i (c+d x)}\right )-(e+f x)^2 \log \left (1+e^{i (c+d x)}\right )+\frac {2 i f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )+i f \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )\right )}{d^2}+\frac {2 f \left (-i d (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )+f \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )\right )}{d^2}}{a d}+\frac {i \left (a^2-b^2\right ) \left (-2 \sqrt {a^2-b^2} d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )+2 \sqrt {a^2-b^2} d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )-i \left (d^2 \left (2 \sqrt {-a^2+b^2} e^2 \arctan \left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+\sqrt {a^2-b^2} f x (2 e+f x) \left (\log \left (1-\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-\log \left (1+\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )\right )+2 \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,\frac {b e^{i (c+d x)}}{-i a+\sqrt {-a^2+b^2}}\right )-2 \sqrt {a^2-b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{i (c+d x)}}{i a+\sqrt {-a^2+b^2}}\right )\right )\right )}{a b \sqrt {-\left (a^2-b^2\right )^2} d^3} \] Input:

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

Output:

-1/3*(x*(3*e^2 + 3*e*f*x + f^2*x^2))/b + ((e + f*x)^2*Log[1 - E^(I*(c + d* 
x))] - (e + f*x)^2*Log[1 + E^(I*(c + d*x))] + ((2*I)*f*(d*(e + f*x)*PolyLo 
g[2, -E^(I*(c + d*x))] + I*f*PolyLog[3, -E^(I*(c + d*x))]))/d^2 + (2*f*((- 
I)*d*(e + f*x)*PolyLog[2, E^(I*(c + d*x))] + f*PolyLog[3, E^(I*(c + d*x))] 
))/d^2)/(a*d) + (I*(a^2 - b^2)*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*PolyLog[2 
, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b^2]*d*f 
*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2]))] - I 
*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - 
b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)))/((-I) 
*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b 
^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I)*a + S 
qrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + d*x))) 
/(I*a + Sqrt[-a^2 + b^2]))])))/(a*b*Sqrt[-(a^2 - b^2)^2]*d^3)
 

Rubi [A] (verified)

Time = 4.11 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.13, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5054, 4908, 3042, 3777, 3042, 3777, 25, 3042, 3118, 4671, 3011, 2720, 5036, 17, 3042, 3777, 3042, 3777, 25, 3042, 3118, 3804, 2694, 27, 2620, 3011, 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.

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

\(\Big \downarrow \) 5054

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5036

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((-2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/d + ((e + f*x)^2*Cos[c + d*x])/ 
d + (2*f*((I*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/d - (f*PolyLog[3, -E^ 
(I*(c + d*x))])/d^2))/d - (2*f*((I*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/ 
d - (f*PolyLog[3, E^(I*(c + d*x))])/d^2))/d - (2*f*((f*Cos[c + d*x])/d^2 + 
 ((e + f*x)*Sin[c + d*x])/d))/d)/a - (b*((a*(e + f*x)^3)/(3*b^2*f) - (2*(a 
^2 - b^2)*(((-1/2*I)*b*(((e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sq 
rt[a^2 - b^2])])/(b*d) - (2*f*((I*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x) 
))/(a - Sqrt[a^2 - b^2])])/d - (f*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sq 
rt[a^2 - b^2])])/d^2))/(b*d)))/Sqrt[a^2 - b^2] + ((I/2)*b*(((e + f*x)^2*Lo 
g[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (2*f*((I*(e + 
f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d - (f*PolyL 
og[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d^2))/(b*d)))/Sqrt[a^2 
 - b^2]))/b^2 - (-(((e + f*x)^2*Cos[c + d*x])/d) + (2*f*((f*Cos[c + d*x])/ 
d^2 + ((e + f*x)*Sin[c + d*x])/d))/d)/b))/a
 

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 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 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 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 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 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]
 

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]
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*cos(d*x+c)*cot(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. 2109 vs. \(2 (481) = 962\).

Time = 0.32 (sec) , antiderivative size = 2109, normalized size of antiderivative = 3.79 \[ \int \frac {(e+f x)^2 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(2*a*d^3*f^2*x^3 + 6*a*d^3*e*f*x^2 + 6*a*d^3*e^2*x - 6*b*f^2*sqrt(-(a 
^2 - b^2)/b^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) + 6*b*f^2*sqrt(-(a^2 
- b^2)/b^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) - 6*b*f^2*sqrt(-(a^2 - b 
^2)/b^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) + 6*b*f^2*sqrt(-(a^2 - b^2 
)/b^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) - 6*b*f^2*polylog(3, cos(d*x 
 + c) + I*sin(d*x + c)) - 6*b*f^2*polylog(3, cos(d*x + c) - I*sin(d*x + c) 
) + 6*b*f^2*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) + 6*b*f^2*polylog(3 
, -cos(d*x + c) - I*sin(d*x + c)) + 6*(-I*b*d*f^2*x - I*b*d*e*f)*sqrt(-(a^ 
2 - b^2)/b^2)*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) + 6*(I*b*d*f^2*x + I 
*b*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*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) + 
 6*(I*b*d*f^2*x + I*b*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*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) + 6*(-I*b*d*f^2*x - I*b*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*d 
ilog((-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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