\(\int \frac {(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) [334]

Optimal result

Integrand size = 34, antiderivative size = 825 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^3}{3 b^3 f}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 f^2 \cos (c+d x)}{a d^3}-\frac {2 \left (a^2-b^2\right ) f^2 \cos (c+d x)}{a b^2 d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {\left (a^2-b^2\right ) (e+f x)^2 \cos (c+d x)}{a b^2 d}-\frac {f (e+f x) \cos ^2(c+d x)}{2 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {2 \left (a^2-b^2\right )^{3/2} f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^3}-\frac {2 i \left (a^2-b^2\right )^{3/2} f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}-\frac {2 \left (a^2-b^2\right ) f (e+f x) \sin (c+d x)}{a b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d} \] Output:

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

Mathematica [A] (verified)

Time = 5.63 (sec) , antiderivative size = 1254, normalized size of antiderivative = 1.52 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/24*(-24*a^3*d^3*e^2*x + 36*a*b^2*d^3*e^2*x - 24*a^3*d^3*e*f*x^2 + 36*a* 
b^2*d^3*e*f*x^2 - 8*a^3*d^3*f^2*x^3 + 12*a*b^2*d^3*f^2*x^3 + 48*(a^2 - b^2 
)^(3/2)*d^2*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] - 24*a^2 
*b*d^2*e^2*Cos[c + d*x] + 48*a^2*b*f^2*Cos[c + d*x] - 48*a^2*b*d^2*e*f*x*C 
os[c + d*x] - 24*a^2*b*d^2*f^2*x^2*Cos[c + d*x] + 6*a*b^2*d*e*f*Cos[2*(c + 
 d*x)] + 6*a*b^2*d*f^2*x*Cos[2*(c + d*x)] - 24*b^3*d^2*e^2*Log[1 - E^(I*(c 
 + d*x))] - 48*b^3*d^2*e*f*x*Log[1 - E^(I*(c + d*x))] - 24*b^3*d^2*f^2*x^2 
*Log[1 - E^(I*(c + d*x))] + 24*b^3*d^2*e^2*Log[1 + E^(I*(c + d*x))] + 48*b 
^3*d^2*e*f*x*Log[1 + E^(I*(c + d*x))] + 24*b^3*d^2*f^2*x^2*Log[1 + E^(I*(c 
 + d*x))] - (48*I)*(a^2 - b^2)^(3/2)*d^2*e*f*x*Log[1 + (I*b*E^(I*(c + d*x) 
))/(-a + Sqrt[a^2 - b^2])] - (24*I)*(a^2 - b^2)^(3/2)*d^2*f^2*x^2*Log[1 + 
(I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + (48*I)*(a^2 - b^2)^(3/2)*d 
^2*e*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (24*I)*(a^ 
2 - b^2)^(3/2)*d^2*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b 
^2])] - (48*I)*b^3*d*e*f*PolyLog[2, -E^(I*(c + d*x))] - (48*I)*b^3*d*f^2*x 
*PolyLog[2, -E^(I*(c + d*x))] + (48*I)*b^3*d*e*f*PolyLog[2, E^(I*(c + d*x) 
)] + (48*I)*b^3*d*f^2*x*PolyLog[2, E^(I*(c + d*x))] - 48*(a^2 - b^2)^(3/2) 
*d*e*f*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 48*(a 
^2 - b^2)^(3/2)*d*f^2*x*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 
 - b^2])] + 48*(a^2 - b^2)^(3/2)*d*e*f*PolyLog[2, (I*b*E^(I*(c + d*x)))...
 

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.

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

\(\Big \downarrow \) 5054

\(\displaystyle \frac {\int (e+f x)^2 \cos ^3(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx-\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4905

\(\displaystyle \frac {-\frac {2 f \int (e+f x) \cos ^3(c+d x)dx}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {2 f \left (\frac {2}{3} \int (e+f x) \cos (c+d x)dx+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 f \left (\frac {2}{3} \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {2 f \left (\frac {2}{3} \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 f \left (\frac {2}{3} \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\int (e+f x)^2 \cos (c+d x) \cot (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {-\int (e+f x)^2 \sin (c+d x)dx+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\int (e+f x)^2 \sin (c+d x)dx+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {2 f \int (e+f x) \cos (c+d x)dx}{d}+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {2 f \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}+\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)dx-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {2 f \int (e+f x) \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {2 f \int (e+f x) \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 5036

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x)^2 \cos ^2(c+d x)dx}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (-\frac {f^2 \int \cos ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}\right )}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (-\frac {f^2 \int \cos ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (-\frac {f^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{2 d^2}+\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (-\frac {f^2 \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\int (e+f x)^2 \cos ^2(c+d x) \sin (c+d x)dx}{b}+\frac {a \left (\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 4905

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {2 f \int (e+f x) \cos ^3(c+d x)dx}{3 d}-\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{b}+\frac {a \left (\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {2 f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{3 d}-\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}}{b}+\frac {a \left (\frac {f (e+f x) \cos ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{2 d^2}+\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{b^2}\right )}{a}+\frac {\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}-\frac {2 f \left (\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\frac {f \cos ^3(c+d x)}{9 d^2}+\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )}{3 d}-\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}+\frac {(e+f x)^2 \cos ^3(c+d x)}{3 d}+\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

Input:

Int[((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x])/(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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

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 3115

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

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 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 3792

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

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 4905

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

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

Input:

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

Output:

int((f*x+e)^2*cos(d*x+c)^3*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. 2787 vs. \(2 (739) = 1478\).

Time = 0.41 (sec) , antiderivative size = 2787, normalized size of antiderivative = 3.38 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(2*(2*a^3 - 3*a*b^2)*d^3*f^2*x^3 + 6*(2*a^3 - 3*a*b^2)*d^3*e*f*x^2 + 
12*b^3*f^2*polylog(3, cos(d*x + c) + I*sin(d*x + c)) + 12*b^3*f^2*polylog( 
3, cos(d*x + c) - I*sin(d*x + c)) - 12*b^3*f^2*polylog(3, -cos(d*x + c) + 
I*sin(d*x + c)) - 12*b^3*f^2*polylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 
12*(a^2*b - b^3)*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) + 12*(a^2*b - b^3)*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) - 12*(a^2*b - b^3)*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) + 12*(a^2*b - b^3)*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*s 
in(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 6*(a*b^2*d*f^2*x + a*b^2*d*e*f)* 
cos(d*x + c)^2 - 12*(I*(a^2*b - b^3)*d*f^2*x + I*(a^2*b - b^3)*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) - 12*(-I*(a^2*b 
 - b^3)*d*f^2*x - I*(a^2*b - b^3)*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) - 12*(-I*(a^2*b - b^3)*d*f^2*x - I*(a^2*b - 
b^3)*d*e*f)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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