Optimal. Leaf size=43 \[ -2 i e^{2 i a} x+2 i e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac {i x^3}{3} \]
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Rubi [F] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \cot (a+i \log (x)) \, dx &=\int x^2 \cot (a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.02, 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.
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fricas [B] time = 1.48, size = 82, normalized size = 1.91 \[ -\frac {1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} - \sqrt {-e^{\left (6 i \, a\right )}} \log \left (\frac {1}{2} \, {\left (2 \, x e^{\left (2 i \, a\right )} + 2 i \, \sqrt {-e^{\left (6 i \, a\right )}}\right )} e^{\left (-2 i \, a\right )}\right ) + \sqrt {-e^{\left (6 i \, a\right )}} \log \left (\frac {1}{2} \, {\left (2 \, x e^{\left (2 i \, a\right )} - 2 i \, \sqrt {-e^{\left (6 i \, a\right )}}\right )} e^{\left (-2 i \, a\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 47, normalized size = 1.09 \[ -\frac {1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} + i \, e^{\left (3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 33, normalized size = 0.77 \[ -\frac {i x^{3}}{3}-2 i {\mathrm e}^{2 i a} x +2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 130, normalized size = 3.02 \[ -\frac {1}{3} i \, x^{3} + 2 \, x {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - \frac {1}{6} \, {\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{6} \, {\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) + \frac {1}{2} \, {\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 40, normalized size = 0.93 \[ -\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,{\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,2{}\mathrm {i}-\frac {x^3\,1{}\mathrm {i}}{3}-x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 63, normalized size = 1.47 \[ - \frac {i x^{3}}{3} - 2 i x e^{2 i a} - \left (i \log {\left (x e^{2 i a} - e^{3 i a} \right )} - i \log {\left (x e^{2 i a} + e^{3 i a} \right )}\right ) e^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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