Integrand size = 16, antiderivative size = 302 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5} \]
-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
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1632\) vs. \(2(302)=604\).
Time = 7.09 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.40 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx =\text {Too large to display} \]
-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...
Time = 2.19 (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 |
\(\displaystyle -2 i \left (\frac {2 i d \left (\frac {i (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {3 i d \left (\frac {i d \left (\frac {d \operatorname {PolyLog}\left (5,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{2 b}\right )}{b}-\frac {i (c+d x)^4 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {2 d \left (-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^3 \cot (a+b x)}{b}-\frac {(c+d x)^4}{4 d}\right )}{b}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}+\frac {i (c+d x)^5}{5 d}\) |
((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)
3.2.78.3.1 Defintions of rubi rules used
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]
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]
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]]]
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]
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]
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]
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]
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]
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 = 1.38 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.22
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*ex p(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-I*c^4*x-1/5*I/d*c^5-12*I/b^2 *d^3*c*x^2-12*I/b^4*d^3*c*a^2-12*I/b^4*d^3*c*polylog(2,exp(I*(b*x+a)))-12* I/b^4*d^3*c*polylog(2,-exp(I*(b*x+a)))+4*I/b^2*d^4*polylog(2,exp(I*(b*x+a) ))*x^3-24*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x-24*I/b^4*d^4*polylog(4,-ex p(I*(b*x+a)))*x-2*I/b^4*d^4*a^4*x-12*I/b^4*d^4*polylog(2,exp(I*(b*x+a)))*x -8*I/b^3*c^2*d^2*a^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)))+4*I/b^2*d^4*polylog(2,-exp(I *(b*x+a)))*x^3-4/b*d^3*c*ln(1-exp(I*(b*x+a)))*x^3-4/b*d*c^3*ln(1-exp(I*(b* x+a)))*x-4/b*d*c^3*ln(exp(I*(b*x+a))+1)*x-4/b^2*d*c^3*ln(1-exp(I*(b*x+a))) *a+12/b^3*a^2*c^2*d^2*ln(exp(I*(b*x+a)))-6/b^3*a^2*c^2*d^2*ln(exp(I*(b*x+a ))-1)+6/b^3*c^2*d^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/b^3*d^3*c*polylog(3,-exp(I*(b*x+a)))*x+12/b^3*d^3*c*ln(1-ex p(I*(b*x+a)))*x+12/b^3*d^3*c*ln(exp(I*(b*x+a))+1)*x-4/b*d^3*c*ln(exp(I*(b* x+a))+1)*x^3-6/b*c^2*d^2*ln(1-exp(I*(b*x+a)))*x^2-6/b*c^2*d^2*ln(exp(I*(b* x+a))+1)*x^2-12*I/b^4*d^4*polylog(2,-exp(I*(b*x+a)))*x+12*I/b^4*d^4*a^2*x+ 6*I/b^4*d^3*c*a^4-24*I/b^4*d^3*c*polylog(4,exp(I*(b*x+a)))-24*I/b^4*d^3...
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 = 0.30 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.80 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \]
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*(...
\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int \left (c + d x\right )^{4} \cot ^{3}{\left (a + b x \right )}\, dx \]
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 = 3.68 (sec) , antiderivative size = 7158, normalized size of antiderivative = 23.70 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \]
-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*(...
\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int {\mathrm {cot}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^4 \,d x \]