Optimal. Leaf size=64 \[ -\frac{2 e^{2 i a} x^3}{-x^2+e^{2 i a}}-6 e^{2 i a} x+6 e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{x^3}{3} \]
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Rubi [F] time = 0.0509217, 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 = {} \[ \int x^2 \cot ^2(a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^2 \cot ^2(a+i \log (x)) \, dx &=\int x^2 \cot ^2(a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.126752, size = 100, normalized size = 1.56 \[ \frac{2 x (\cos (3 a)+i \sin (3 a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-4 i x \sin (2 a)-4 x \cos (2 a)+6 \cos (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+6 i \sin (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 75, normalized size = 1.2 \begin{align*} -{\frac{7\,{x}^{3}}{3}}-2\,{\frac{{x}^{3}}{ \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1}}-6\, \left ({{\rm e}^{ia}} \right ) ^{2}x-3\, \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}-x \right ) +3\, \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23497, size = 475, normalized size = 7.42 \begin{align*} -\frac{2 \, x^{5} + x^{3}{\left (22 \, \cos \left (2 \, a\right ) + 22 i \, \sin \left (2 \, a\right )\right )} + 18 \,{\left ({\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{2} - x{\left (36 \, \cos \left (4 \, a\right ) + 36 i \, \sin \left (4 \, a\right )\right )} +{\left (18 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) -{\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (18 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) -{\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) -{\left (x^{2}{\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} -{\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - 9 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\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 ) +{\left (x^{2}{\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} -{\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) + 9 \,{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\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 )}{6 \, x^{2} - 6 \, \cos \left (2 \, a\right ) - 6 i \, \sin \left (2 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x^{3} -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 7 \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.949005, size = 60, normalized size = 0.94 \begin{align*} - \frac{x^{3}}{3} - 4 x e^{2 i a} + \frac{2 x e^{4 i a}}{x^{2} - e^{2 i a}} - 3 \left (\log{\left (x - e^{i a} \right )} - \log{\left (x + e^{i a} \right )}\right ) e^{3 i a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31541, size = 112, normalized size = 1.75 \begin{align*} -\frac{x^{5}}{3 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac{11 \, x^{3} e^{\left (2 i \, a\right )}}{3 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac{6 \, \arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right ) e^{\left (4 i \, a\right )}}{\sqrt{-e^{\left (2 i \, a\right )}}} + \frac{10 \, x e^{\left (4 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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