\(\int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx\) [164]

Optimal result

Integrand size = 22, antiderivative size = 299 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {3 d^2 (c+d x)^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\frac {i (c+d x)^5}{5 d}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}+\frac {3 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b} \] Output:

-3/4*d^2*(d*x+c)^2/b^3+1/4*(d*x+c)^4/b-1/5*I*(d*x+c)^5/d+(d*x+c)^4*ln(1-ex 
p(2*I*(b*x+a)))/b-2*I*d*(d*x+c)^3*polylog(2,exp(2*I*(b*x+a)))/b^2+3*d^2*(d 
*x+c)^2*polylog(3,exp(2*I*(b*x+a)))/b^3+3*I*d^3*(d*x+c)*polylog(4,exp(2*I* 
(b*x+a)))/b^4-3/2*d^4*polylog(5,exp(2*I*(b*x+a)))/b^5+3/2*d^3*(d*x+c)*cos( 
b*x+a)*sin(b*x+a)/b^4-d*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)/b^2-3/4*d^4*sin(b* 
x+a)^2/b^5+3/2*d^2*(d*x+c)^2*sin(b*x+a)^2/b^3-1/2*(d*x+c)^4*sin(b*x+a)^2/b
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2918\) vs. \(2(299)=598\).

Time = 6.29 (sec) , antiderivative size = 2918, normalized size of antiderivative = 9.76 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Result too large to show} \] Input:

Integrate[(c + d*x)^4*Cos[a + b*x]^2*Cot[a + b*x],x]
 

Output:

-((c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2 
*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2 
*Log[1 + E^((-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, -E^((-I 
)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, E^((-I)*(a + b*x))] + 
(6*I)*(1 - E^((-2*I)*a))*PolyLog[3, -E^((-I)*(a + b*x))] + (6*I)*(1 - E^(( 
-2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))]))/b^3) - (c*d^3*E^(I*a)*Csc[a]*(( 
b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a 
 + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - 
 6*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - 6*b^2*(1 - 
 E^((-2*I)*a))*x^2*PolyLog[2, E^((-I)*(a + b*x))] + (12*I)*b*(1 - E^((-2*I 
)*a))*x*PolyLog[3, -E^((-I)*(a + b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*Po 
lyLog[3, E^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*PolyLog[4, -E^((-I)*( 
a + b*x))] + 12*(1 - E^((-2*I)*a))*PolyLog[4, E^((-I)*(a + b*x))]))/b^4 - 
(d^4*E^(I*a)*Csc[a]*((2*b^5*x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a) 
)*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 
 + E^((-I)*(a + b*x))] - 20*b^3*(1 - E^((-2*I)*a))*x^3*PolyLog[2, -E^((-I) 
*(a + b*x))] - 20*b^3*(1 - E^((-2*I)*a))*x^3*PolyLog[2, E^((-I)*(a + b*x)) 
] + (60*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, -E^((-I)*(a + b*x))] + (6 
0*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, E^((-I)*(a + b*x))] + 120*b*(1 
- E^((-2*I)*a))*x*PolyLog[4, -E^((-I)*(a + b*x))] + 120*b*(1 - E^((-2*I...
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {4908, 3042, 25, 4202, 2620, 3011, 4904, 3042, 3792, 17, 3042, 3791, 17, 7163, 7163, 2720, 7143}

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

\(\displaystyle \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx\)

\(\Big \downarrow \) 4908

\(\displaystyle \int (c+d x)^4 \cot (a+b x)dx-\int (c+d x)^4 \cos (a+b x) \sin (a+b x)dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

\(\displaystyle -\int (c+d x)^4 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx-\int (c+d x)^4 \cos (a+b x) \sin (a+b x)dx\)

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^4}{1+e^{i (2 a+2 b x+\pi )}}dx-\int (c+d x)^4 \cos (a+b x) \sin (a+b x)dx-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (\frac {2 i d \int (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\int (c+d x)^4 \cos (a+b x) \sin (a+b x)dx-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\int (c+d x)^4 \cos (a+b x) \sin (a+b x)dx-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 4904

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \int (c+d x)^3 \sin ^2(a+b x)dx}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \int (c+d x)^3 \sin (a+b x)^2dx}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {2 d \left (-\frac {3 d^2 \int (c+d x) \sin ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^3dx+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}\right )}{b}+2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {2 d \left (-\frac {3 d^2 \int (c+d x) \sin ^2(a+b x)dx}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}+2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 d \left (-\frac {3 d^2 \int (c+d x) \sin (a+b x)^2dx}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}+2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {2 d \left (-\frac {3 d^2 \left (\frac {1}{2} \int (c+d x)dx+\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}+2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 17

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \left (\frac {i d \int (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \left (\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \left (\frac {i d \left (\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {2 d \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )}{b}+2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \left (\frac {i d \left (\frac {d \operatorname {PolyLog}\left (5,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^5}{5 d}\)

Input:

Int[(c + d*x)^4*Cos[a + b*x]^2*Cot[a + b*x],x]
 

Output:

((-1/5*I)*(c + d*x)^5)/d + (2*I)*(((-1/2*I)*(c + d*x)^4*Log[1 + E^(I*(2*a 
+ Pi + 2*b*x))])/b + ((2*I)*d*(((I/2)*(c + d*x)^3*PolyLog[2, -E^(I*(2*a + 
Pi + 2*b*x))])/b - (((3*I)/2)*d*(((-1/2*I)*(c + d*x)^2*PolyLog[3, -E^(I*(2 
*a + Pi + 2*b*x))])/b + (I*d*(((-1/2*I)*(c + d*x)*PolyLog[4, -E^(I*(2*a + 
Pi + 2*b*x))])/b + (d*PolyLog[5, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b)) 
/b))/b) - ((c + d*x)^4*Sin[a + b*x]^2)/(2*b) + (2*d*((c + d*x)^4/(8*d) - ( 
(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (3*d*(c + d*x)^2*Sin[a + b* 
x]^2)/(4*b^2) - (3*d^2*((c + d*x)^2/(4*d) - ((c + d*x)*Cos[a + b*x]*Sin[a 
+ b*x])/(2*b) + (d*Sin[a + b*x]^2)/(4*b^2)))/(2*b^2)))/b
 

Defintions of rubi rules used

rule 17

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

rule 25

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

rule 2620

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

rule 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 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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4904

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

rule 4908

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (270 ) = 540\).

Time = 2.75 (sec) , antiderivative size = 1335, normalized size of antiderivative = 4.46

method	result	size
risch	\(\text {Expression too large to display}\)	\(1335\)

Input:

int((d*x+c)^4*cos(b*x+a)^2*cot(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4/b^4*d*(2*b^2*d^3*x^3+6*b^2*c*d^2*x^2+6*b^2*c^2*d*x+2*b^2*c^3-3*d^3*x- 
3*c*d^2)*sin(2*b*x+2*a)-2*I*d*c^3*x^2-I*d^3*c*x^4-2*I*d^2*c^2*x^3-1/b^5*d^ 
4*ln(1-exp(I*(b*x+a)))*a^4-2/b^5*d^4*a^4*ln(exp(I*(b*x+a)))+1/b^5*d^4*a^4* 
ln(exp(I*(b*x+a))-1)+12/b^3*d^2*c^2*polylog(3,-exp(I*(b*x+a)))+12/b^3*d^2* 
c^2*polylog(3,exp(I*(b*x+a)))+1/b*d^4*ln(exp(I*(b*x+a))+1)*x^4+1/b*d^4*ln( 
1-exp(I*(b*x+a)))*x^4+12/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^2+12/b^3*d^4 
*polylog(3,exp(I*(b*x+a)))*x^2+8/5*I/b^5*d^4*a^5+1/8*(2*b^4*d^4*x^4+8*b^4* 
c*d^3*x^3+12*b^4*c^2*d^2*x^2+8*b^4*c^3*d*x+2*b^4*c^4-6*b^2*d^4*x^2-12*b^2* 
c*d^3*x-6*b^2*c^2*d^2+3*d^4)/b^5*cos(2*b*x+2*a)-12*I/b^2*d^3*c*polylog(2,- 
exp(I*(b*x+a)))*x^2-12*I/b^2*d^3*c*polylog(2,exp(I*(b*x+a)))*x^2-8*I/b^3*d 
^3*c*a^3*x-12*I/b^2*d^2*c^2*polylog(2,-exp(I*(b*x+a)))*x-12*I/b^2*d^2*c^2* 
polylog(2,exp(I*(b*x+a)))*x+12*I/b^2*d^2*c^2*a^2*x-8*I/b*d*c^3*a*x+4/b*d^3 
*c*ln(1-exp(I*(b*x+a)))*x^3+24*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x-4*I/b 
^2*d^4*polylog(2,exp(I*(b*x+a)))*x^3-4*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)) 
)*x^3-4*I/b^2*d*c^3*a^2-4*I/b^2*d*c^3*polylog(2,-exp(I*(b*x+a)))-4*I/b^2*d 
*c^3*polylog(2,exp(I*(b*x+a)))+8*I/b^3*d^2*c^2*a^3+24*I/b^4*d^3*c*polylog( 
4,-exp(I*(b*x+a)))+24*I/b^4*d^3*c*polylog(4,exp(I*(b*x+a)))-6*I/b^4*d^3*c* 
a^4+2*I/b^4*d^4*a^4*x+24*I/b^4*d^4*polylog(4,-exp(I*(b*x+a)))*x+4/b^4*d^3* 
c*ln(1-exp(I*(b*x+a)))*a^3+4/b^2*d*c^3*ln(1-exp(I*(b*x+a)))*a-6/b^3*d^2*c^ 
2*ln(1-exp(I*(b*x+a)))*a^2+24/b^3*d^3*c*polylog(3,-exp(I*(b*x+a)))*x+24...
 

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. 1453 vs. \(2 (266) = 532\).

Time = 0.17 (sec) , antiderivative size = 1453, normalized size of antiderivative = 4.86 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cos(b*x+a)^2*cot(b*x+a),x, algorithm="fricas")
 

Output:

-1/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 48*d^4*polylog(5, cos(b*x + a) + I*s 
in(b*x + a)) + 48*d^4*polylog(5, cos(b*x + a) - I*sin(b*x + a)) + 48*d^4*p 
olylog(5, -cos(b*x + a) + I*sin(b*x + a)) + 48*d^4*polylog(5, -cos(b*x + a 
) - I*sin(b*x + a)) + 3*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 - (2*b^4*d^4*x^4 + 8 
*b^4*c*d^3*x^3 + 2*b^4*c^4 - 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^ 
2*d^4)*x^2 + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^2 + 2*(2*b^3*d^ 
4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d 
^4)*x)*cos(b*x + a)*sin(b*x + a) + 2*(2*b^4*c^3*d - 3*b^2*c*d^3)*x + 8*(I* 
b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + 3*I*b^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(c 
os(b*x + a) + I*sin(b*x + a)) + 8*(-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3* 
I*b^3*c^2*d^2*x - I*b^3*c^3*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) + 8*(- 
I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3*I*b^3*c^2*d^2*x - I*b^3*c^3*d)*dilog 
(-cos(b*x + a) + I*sin(b*x + a)) + 8*(I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + 
3*I*b^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(-cos(b*x + a) - I*sin(b*x + a)) - 2 
*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4* 
c^4)*log(cos(b*x + a) + I*sin(b*x + a) + 1) - 2*(b^4*d^4*x^4 + 4*b^4*c*d^3 
*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*log(cos(b*x + a) - I*s 
in(b*x + a) + 1) - 2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3* 
b*c*d^3 + a^4*d^4)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - 2*( 
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*...
 

Sympy [F]

\[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\int \left (c + d x\right )^{4} \cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**4*cos(b*x+a)**2*cot(b*x+a),x)
 

Output:

Integral((c + d*x)**4*cos(a + b*x)**2*cot(a + b*x), x)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1654 vs. \(2 (266) = 532\).

Time = 0.28 (sec) , antiderivative size = 1654, normalized size of antiderivative = 5.53 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cos(b*x+a)^2*cot(b*x+a),x, algorithm="maxima")
 

Output:

-1/40*(20*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*c^4 - 80*(sin(b*x + a)^2 
- log(sin(b*x + a)^2))*a*c^3*d/b + 120*(sin(b*x + a)^2 - log(sin(b*x + a)^ 
2))*a^2*c^2*d^2/b^2 - 80*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^3*c*d^3/ 
b^3 + 20*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^4*d^4/b^4 - (-8*I*(b*x + 
 a)^5*d^4 - 40*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^4 - 960*d^4*polylog(5, -e^( 
I*b*x + I*a)) - 960*d^4*polylog(5, e^(I*b*x + I*a)) - 80*(I*b^2*c^2*d^2 - 
2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^3 - 80*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d 
^2 + 3*I*a^2*b*c*d^3 - I*a^3*d^4)*(b*x + a)^2 - 40*(-I*(b*x + a)^4*d^4 + 4 
*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - 
I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c 
*d^3 + I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 40* 
(I*(b*x + a)^4*d^4 + 4*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^3 + 6*(I*b^2*c^2*d^ 
2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^2 + 4*(I*b^3*c^3*d - 3*I*a*b^2*c^ 
2*d^2 + 3*I*a^2*b*c*d^3 - I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), -cos 
(b*x + a) + 1) + 5*(2*(b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 3*( 
2*a^2 - 1)*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(2*b^2*c^2*d^2 - 4*a* 
b*c*d^3 + (2*a^2 - 1)*d^4)*(b*x + a)^2 + 4*(2*b^3*c^3*d - 6*a*b^2*c^2*d^2 
+ 3*(2*a^2 - 1)*b*c*d^3 - (2*a^3 - 3*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - 
 160*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^ 
4 - I*a^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 ...
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cos \left (b x + a\right )^{2} \cot \left (b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^4*cos(b*x+a)^2*cot(b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*cos(b*x + a)^2*cot(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4 \,d x \] Input:

int(cos(a + b*x)^2*cot(a + b*x)*(c + d*x)^4,x)
 

Output:

int(cos(a + b*x)^2*cot(a + b*x)*(c + d*x)^4, x)
 

Reduce [F]

\[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {2 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x^{4}d x \right ) b \,d^{4}+8 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x^{3}d x \right ) b c \,d^{3}+12 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x^{2}d x \right ) b \,c^{2} d^{2}+8 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x d x \right ) b \,c^{3} d -2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) c^{4}+2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c^{4}-\sin \left (b x +a \right )^{2} c^{4}}{2 b} \] Input:

int((d*x+c)^4*cos(b*x+a)^2*cot(b*x+a),x)
 

Output:

(2*int(cos(a + b*x)**2*cot(a + b*x)*x**4,x)*b*d**4 + 8*int(cos(a + b*x)**2 
*cot(a + b*x)*x**3,x)*b*c*d**3 + 12*int(cos(a + b*x)**2*cot(a + b*x)*x**2, 
x)*b*c**2*d**2 + 8*int(cos(a + b*x)**2*cot(a + b*x)*x,x)*b*c**3*d - 2*log( 
tan((a + b*x)/2)**2 + 1)*c**4 + 2*log(tan((a + b*x)/2))*c**4 - sin(a + b*x 
)**2*c**4)/(2*b)