Optimal. Leaf size=43 \[ -2 i e^{2 i a} x+2 i e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i x^3}{3} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.0223548, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int x^2 \cot (a+i \log (x)) \, dx &=\int x^2 \cot (a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.0181679, size = 66, normalized size = 1.53 \[ 2 x \sin (2 a)-2 i x \cos (2 a)+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i x^3}{3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.089, size = 62, normalized size = 1.4 \begin{align*}{\frac{i}{3}}{x}^{3}+i \left ( -{\frac{2\,{x}^{3}}{3}}-2\, \left ({{\rm e}^{ia}} \right ) ^{2}x+ \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}+x \right ) - \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}-x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.18713, size = 176, normalized size = 4.09 \begin{align*} -\frac{1}{3} i \, x^{3} + 2 \, x{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - \frac{1}{6} \,{\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - \frac{1}{6} \,{\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + \frac{1}{2} \,{\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + \frac{1}{2} \,{\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{i \, x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.497822, size = 46, normalized size = 1.07 \begin{align*} - \frac{i x^{3}}{3} - 2 i x e^{2 i a} - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{3 i a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.46495, size = 63, normalized size = 1.47 \begin{align*} -\frac{1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} + i \, e^{\left (3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]