3.2.79 \(\int (c+d x)^3 \cot ^3(a+b x) \, dx\) [179]

3.2.79.1 Optimal result

Integrand size = 16, antiderivative size = 256 \[ \int (c+d x)^3 \cot ^3(a+b x) \, dx=-\frac {3 i d (c+d x)^2}{2 b^2}-\frac {(c+d x)^3}{2 b}+\frac {i (c+d x)^4}{4 d}-\frac {3 d (c+d x)^2 \cot (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot ^2(a+b x)}{2 b}+\frac {3 d^2 (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i d^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \]

output

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

3.2.79.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1032\) vs. \(2(256)=512\).

Time = 7.00 (sec) , antiderivative size = 1032, normalized size of antiderivative = 4.03 \[ \int (c+d x)^3 \cot ^3(a+b x) \, dx=-\frac {1}{4} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cot (a)-\frac {(c+d x)^3 \csc ^2(a+b x)}{2 b}+\frac {c d^2 e^{i a} \csc (a) \left (2 b^3 e^{-2 i a} x^3+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1-e^{-i (a+b x)}\right )+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1+e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right )}{2 b^3}+\frac {d^3 e^{i a} \csc (a) \left (b^4 e^{-2 i a} x^4+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1-e^{-i (a+b x)}\right )+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1+e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )\right )}{4 b^4}-\frac {c^3 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 c d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {3 \csc (a) \csc (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2}+\frac {3 c^2 d \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{2 b^2 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {3 d^3 \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{2 b^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]

input

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

output

-1/4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cot[a]) - ((c + d*x)^3 
*Csc[a + b*x]^2)/(2*b) + (c*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))]))/(2*b^3) + 
(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))]))/(4*b^4) - (c^3*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] 
 + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) + (3*c*d^2*Csc[a]*( 
-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[ 
a]^2 + Sin[a]^2)) + (3*Csc[a]*Csc[a + b*x]*(c^2*d*Sin[b*x] + 2*c*d^2*x*Sin 
[b*x] + d^3*x^2*Sin[b*x]))/(2*b^2) + (3*c^2*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcT 
an[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*...
 

3.2.79.3 Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.25, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.312, Rules used = {3042, 25, 4203, 25, 3042, 25, 4202, 2620, 3011, 4203, 17, 25, 3042, 25, 4202, 2620, 2715, 2838, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.2.79.3.1 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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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]
 

3.2.79.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (221 ) = 442\).

Time = 1.33 (sec) , antiderivative size = 1203, normalized size of antiderivative = 4.70

method	result	size
risch	\(\text {Expression too large to display}\)	\(1203\)

input

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

output

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

3.2.79.5 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. 1139 vs. \(2 (214) = 428\).

Time = 0.27 (sec) , antiderivative size = 1139, normalized size of antiderivative = 4.45 \[ \int (c+d x)^3 \cot ^3(a+b x) \, dx=\text {Too large to display} \]

input

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

output

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

3.2.79.6 Sympy [F]

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

input

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

output

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

3.2.79.7 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. 3958 vs. \(2 (214) = 428\).

Time = 1.22 (sec) , antiderivative size = 3958, normalized size of antiderivative = 15.46 \[ \int (c+d x)^3 \cot ^3(a+b x) \, dx=\text {Too large to display} \]

input

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

output

-1/2*(c^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) - 3*a*c^2*d*(1/sin(b*x 
+ a)^2 + log(sin(b*x + a)^2))/b + 3*a^2*c*d^2*(1/sin(b*x + a)^2 + log(sin( 
b*x + a)^2))/b^2 - a^3*d^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/b^3 - 
2*((b*x + a)^4*d^3 + 12*b^2*c^2*d - 24*a*b*c*d^2 + 12*a^2*d^3 + 4*(b*c*d^2 
 - a*d^3)*(b*x + a)^3 + 6*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a)^2 
- 4*((b*x + a)^3*d^3 - 3*b*c*d^2 + 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a) 
^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 - 1)*d^3)*(b*x + a) + ((b*x + a)^3* 
d^3 - 3*b*c*d^2 + 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d 
 - 2*a*b*c*d^2 + (a^2 - 1)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 2*((b*x + a) 
^3*d^3 - 3*b*c*d^2 + 3*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^ 
2*d - 2*a*b*c*d^2 + (a^2 - 1)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x 
+ a)^3*d^3 + 3*I*b*c*d^2 - 3*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^ 
2 + 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 + (-I*a^2 + I)*d^3)*(b*x + a))*sin(4*b 
*x + 4*a) - 2*(I*(b*x + a)^3*d^3 - 3*I*b*c*d^2 + 3*I*a*d^3 + 3*(I*b*c*d^2 
- I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + (I*a^2 - I)*d^3) 
*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 12 
*(b*c*d^2 - a*d^3 + (b*c*d^2 - a*d^3)*cos(4*b*x + 4*a) - 2*(b*c*d^2 - a*d^ 
3)*cos(2*b*x + 2*a) + (I*b*c*d^2 - I*a*d^3)*sin(4*b*x + 4*a) + 2*(-I*b*c*d 
^2 + I*a*d^3)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 
4*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2...
 

3.2.79.8 Giac [F]

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

input

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

output

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

3.2.79.9 Mupad [F(-1)]

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

input

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

output

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