\(\int (c+d x)^4 \cot ^3(a+b x) \, dx\) [178]

Optimal result

Integrand size = 16, antiderivative size = 302 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5} \] Output:

-2*I*d*(d*x+c)^3/b^2-1/2*(d*x+c)^4/b+1/5*I*(d*x+c)^5/d-2*d*(d*x+c)^3*cot(b 
*x+a)/b^2-1/2*(d*x+c)^4*cot(b*x+a)^2/b+6*d^2*(d*x+c)^2*ln(1-exp(2*I*(b*x+a 
)))/b^3-(d*x+c)^4*ln(1-exp(2*I*(b*x+a)))/b-6*I*d^3*(d*x+c)*polylog(2,exp(2 
*I*(b*x+a)))/b^4+2*I*d*(d*x+c)^3*polylog(2,exp(2*I*(b*x+a)))/b^2+3*d^4*pol 
ylog(3,exp(2*I*(b*x+a)))/b^5-3*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
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1632\) vs. \(2(302)=604\).

Time = 6.70 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.40 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx =\text {Too large to display} \] Input:

Integrate[(c + d*x)^4*Cot[a + b*x]^3,x]
 

Output:

-1/5*(x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)*Cot[ 
a]) - ((c + d*x)^4*Csc[a + b*x]^2)/(2*b) + (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 - (d^4*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^5 + (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*PolyLog[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...
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.35, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.438, Rules used = {3042, 25, 4203, 25, 3042, 25, 4202, 2620, 3011, 4203, 17, 25, 3042, 25, 4202, 2620, 3011, 2720, 7143, 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.

\(\displaystyle \int (c+d x)^4 \cot ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -(c+d x)^4 \tan \left (a+b x+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (c+d x)^4 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^3dx\)

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 d \int (c+d x)^3 \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}+\int (c+d x)^4 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(c + d*x)^4*Cot[a + b*x]^3,x]
 

Output:

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

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 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 4203

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si 
mp[b*d*(m/(f*(n - 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] 
, x] - Simp[b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free 
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 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. 1876 vs. \(2 (273 ) = 546\).

Time = 0.47 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(1877\)

Input:

int((d*x+c)^4*cot(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

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. 1751 vs. \(2 (266) = 532\).

Time = 0.12 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.80 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="fricas")
 

Output:

1/4*(4*b^4*d^4*x^4 + 16*b^4*c*d^3*x^3 + 24*b^4*c^2*d^2*x^2 + 16*b^4*c^3*d* 
x + 4*b^4*c^4 - 4*(I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 3*I*b 
*c*d^3 + 3*I*(b^3*c^2*d^2 - b*d^4)*x + (-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 
 - I*b^3*c^3*d + 3*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - b*d^4)*x)*cos(2*b*x + 2* 
a))*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) - 4*(-I*b^3*d^4*x^3 - 3*I 
*b^3*c*d^3*x^2 - I*b^3*c^3*d + 3*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - b*d^4)*x + 
 (I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 3*I*b*c*d^3 + 3*I*(b^3 
*c^2*d^2 - b*d^4)*x)*cos(2*b*x + 2*a))*dilog(cos(2*b*x + 2*a) - I*sin(2*b* 
x + 2*a)) + 2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 
- 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4 - (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1 
)*b^2*c^2*d^2 - 4*(a^3 - 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4)*cos(2*b*x + 2*a 
))*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b^4*c^4 
- 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 - 3*a)*b*c*d^3 + (a^4 - 
 6*a^2)*d^4 - (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 
- 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4)*cos(2*b*x + 2*a))*log(-1/2*cos(2*b*x + 
 2*a) - 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 
 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 3*a)*b*c*d^3 - (a^4 - 6*a^2) 
*d^4 + 6*(b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(b^4*c^3*d - 3*b^2*c*d^3)*x - (b^ 
4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 
 3*a)*b*c*d^3 - (a^4 - 6*a^2)*d^4 + 6*(b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(...
 

Sympy [F]

\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int \left (c + d x\right )^{4} \cot ^{3}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**4*cot(b*x+a)**3,x)
                                                                                    
                                                                                    
 

Output:

Integral((c + d*x)**4*cot(a + b*x)**3, 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. 7158 vs. \(2 (266) = 532\).

Time = 3.26 (sec) , antiderivative size = 7158, normalized size of antiderivative = 23.70 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="maxima")
 

Output:

-1/2*(c^4*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) - 4*a*c^3*d*(1/sin(b*x 
+ a)^2 + log(sin(b*x + a)^2))/b + 6*a^2*c^2*d^2*(1/sin(b*x + a)^2 + log(si 
n(b*x + a)^2))/b^2 - 4*a^3*c*d^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/ 
b^3 + a^4*d^4*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/b^4 - 2*(2*(b*x + a 
)^5*d^4 + 40*b^3*c^3*d - 120*a*b^2*c^2*d^2 + 120*a^2*b*c*d^3 - 40*a^3*d^4 
+ 10*(b*c*d^3 - a*d^4)*(b*x + a)^4 + 20*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d 
^4)*(b*x + a)^3 + 20*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^ 
4)*(b*x + a)^2 - 10*((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^ 
2*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + ( 
a^2 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b 
*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a) + ((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 1 
2*a*b*c*d^3 - 6*a^2*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 
 - 2*a*b*c*d^3 + (a^2 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d 
^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 
2*((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*d^4 + 4*(b*c*d^3 
 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 - 1)*d^4)*(b*x 
 + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3* 
a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^4*d^4 + 6*I*b^2*c^2*d^ 
2 - 12*I*a*b*c*d^3 + 6*I*a^2*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 + I)*d^4)*(b*x + a)^2 + 4*(...
 

Giac [F]

\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{3} \,d x } \] Input:

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*cot(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int {\mathrm {cot}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^4 \,d x \] Input:

int(cot(a + b*x)^3*(c + d*x)^4,x)
 

Output:

int(cot(a + b*x)^3*(c + d*x)^4, x)
 

Reduce [F]

\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\frac {4 \left (\int \cot \left (b x +a \right )^{3} x^{4}d x \right ) \sin \left (b x +a \right )^{2} b \,d^{4}+16 \left (\int \cot \left (b x +a \right )^{3} x^{3}d x \right ) \sin \left (b x +a \right )^{2} b c \,d^{3}+24 \left (\int \cot \left (b x +a \right )^{3} x^{2}d x \right ) \sin \left (b x +a \right )^{2} b \,c^{2} d^{2}+16 \left (\int \cot \left (b x +a \right )^{3} x d x \right ) \sin \left (b x +a \right )^{2} b \,c^{3} d +4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right )^{2} c^{4}-4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} c^{4}+\sin \left (b x +a \right )^{2} c^{4}-2 c^{4}}{4 \sin \left (b x +a \right )^{2} b} \] Input:

int((d*x+c)^4*cot(b*x+a)^3,x)
 

Output:

(4*int(cot(a + b*x)**3*x**4,x)*sin(a + b*x)**2*b*d**4 + 16*int(cot(a + b*x 
)**3*x**3,x)*sin(a + b*x)**2*b*c*d**3 + 24*int(cot(a + b*x)**3*x**2,x)*sin 
(a + b*x)**2*b*c**2*d**2 + 16*int(cot(a + b*x)**3*x,x)*sin(a + b*x)**2*b*c 
**3*d + 4*log(tan((a + b*x)/2)**2 + 1)*sin(a + b*x)**2*c**4 - 4*log(tan((a 
 + b*x)/2))*sin(a + b*x)**2*c**4 + sin(a + b*x)**2*c**4 - 2*c**4)/(4*sin(a 
 + b*x)**2*b)