\(\int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [345]

Optimal result

Integrand size = 36, antiderivative size = 1432 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Output:

-3/2*b*f^2*(f*x+e)*polylog(3,exp(2*I*(d*x+c)))/a^2/d^3-6*I*f^2*(f*x+e)*pol 
ylog(2,exp(I*(d*x+c)))/a/d^3-3/4*I*b*f^3*polylog(4,exp(2*I*(d*x+c)))/a^2/d 
^4-3/4*b*f^2*(f*x+e)*sin(d*x+c)^2/a^2/d^3+1/2*(a^2-b^2)*(f*x+e)^3*sin(d*x+ 
c)^2/a^2/b/d-3/8*b*f^3*cos(d*x+c)*sin(d*x+c)/a^2/d^4+6*(a^2-b^2)*f^3*cos(d 
*x+c)/a/b^2/d^4+3/8*(a^2-b^2)*f^3*x/a^2/b/d^3-1/4*I*(a^2-b^2)^2*(f*x+e)^4/ 
a^2/b^3/f+(a^2-b^2)^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2) 
))/a^2/b^3/d+(a^2-b^2)^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1 
/2)))/a^2/b^3/d-3*f*(f*x+e)^2*cos(d*x+c)/a/d^2+6*f^2*(f*x+e)*sin(d*x+c)/a/ 
d^3-6*f*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d^2+6*I*f^2*(f*x+e)*polylog(2, 
-exp(I*(d*x+c)))/a/d^3+3/2*I*b*f*(f*x+e)^2*polylog(2,exp(2*I*(d*x+c)))/a^2 
/d^2+1/4*I*b*(f*x+e)^4/a^2/f-1/4*b*(f*x+e)^3/a^2/d+6*(a^2-b^2)^2*f^2*(f*x+ 
e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d^3+6*(a^2-b^ 
2)^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^3 
/d^3+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2))) 
/a^2/b^3/d^4+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2) 
^(1/2)))/a^2/b^3/d^4+6*f^3*cos(d*x+c)/a/d^4-6*f^3*polylog(3,-exp(I*(d*x+c) 
))/a/d^4+6*f^3*polylog(3,exp(I*(d*x+c)))/a/d^4+6*(a^2-b^2)*f^2*(f*x+e)*sin 
(d*x+c)/a/b^2/d^3-(f*x+e)^3*csc(d*x+c)/a/d-3/4*(a^2-b^2)*f^2*(f*x+e)*sin(d 
*x+c)^2/a^2/b/d^3-3/8*(a^2-b^2)*f^3*cos(d*x+c)*sin(d*x+c)/a^2/b/d^4+3/4*b* 
f*(f*x+e)^2*cos(d*x+c)*sin(d*x+c)/a^2/d^2-3*(a^2-b^2)*f*(f*x+e)^2*cos(d...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4009\) vs. \(2(1432)=2864\).

Time = 13.60 (sec) , antiderivative size = 4009, normalized size of antiderivative = 2.80 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-e^3 - 3*e^2*f*x - 3*e*f^2*x^2 - f^3*x^3)*Csc[c + d*x])/(a*d) - (((-2*I) 
*e^2*(b*d*e - 3*a*f)*x)/d - ((2*I)*e^2*(b*d*e + 3*a*f)*x)/d - (I*b*(e + f* 
x)^4)/((-1 + E^((2*I)*c))*f) + (6*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c 
 + d*x))])/d^2 + (6*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))])/d^2 
 + (2*b*f^3*x^3*Log[1 - E^((-I)*(c + d*x))])/d + (6*e*f*(b*d*e + 2*a*f)*x* 
Log[1 + E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e + a*f)*x^2*Log[1 + E^((-I 
)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 + E^((-I)*(c + d*x))])/d + (2*e^2* 
(b*d*e - 3*a*f)*Log[1 - E^(I*(c + d*x))])/d^2 + (2*e^2*(b*d*e + 3*a*f)*Log 
[1 + E^(I*(c + d*x))])/d^2 + ((6*I)*e*f*(b*d*e + 2*a*f)*PolyLog[2, -E^((-I 
)*(c + d*x))])/d^3 + ((12*I)*f^2*(b*d*e + a*f)*x*PolyLog[2, -E^((-I)*(c + 
d*x))])/d^3 + ((6*I)*b*f^3*x^2*PolyLog[2, -E^((-I)*(c + d*x))])/d^2 + ((6* 
I)*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + ((12*I)*f^2*( 
b*d*e - a*f)*x*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + ((6*I)*b*f^3*x^2*Poly 
Log[2, E^((-I)*(c + d*x))])/d^2 + (12*f^2*(b*d*e + a*f)*PolyLog[3, -E^((-I 
)*(c + d*x))])/d^4 + (12*b*f^3*x*PolyLog[3, -E^((-I)*(c + d*x))])/d^3 + (1 
2*f^2*(b*d*e - a*f)*PolyLog[3, E^((-I)*(c + d*x))])/d^4 + (12*b*f^3*x*Poly 
Log[3, E^((-I)*(c + d*x))])/d^3 - ((12*I)*b*f^3*PolyLog[4, -E^((-I)*(c + d 
*x))])/d^4 - ((12*I)*b*f^3*PolyLog[4, E^((-I)*(c + d*x))])/d^4)/(2*a^2) + 
((a^2 - b^2)^2*((-4*I)*d^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f 
*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f^3*x^4 - (2...
 

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.

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

\(\Big \downarrow \) 5054

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

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 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 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 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]

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

Input:

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

Output:

int((f*x+e)^3*cos(d*x+c)^3*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. 4916 vs. \(2 (1307) = 2614\).

Time = 0.67 (sec) , antiderivative size = 4916, normalized size of antiderivative = 3.43 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [F]

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

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

Output:

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