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 = 0.05, 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 = {} \[ \int x^2 \cot ^2(a+i \log (x)) \, dx \]
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*}
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Mathematica [A] time = 0.13, 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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fricas [B] time = 0.65, size = 102, normalized size = 1.59 \[ -\frac {x^{5} + 11 \, x^{3} e^{\left (2 i \, a\right )} - 9 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )} e^{\left (3 i \, a\right )} \log \left ({\left (x e^{\left (2 i \, a\right )} + e^{\left (3 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}\right ) + 9 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )} e^{\left (3 i \, a\right )} \log \left ({\left (x e^{\left (2 i \, a\right )} - e^{\left (3 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}\right ) - 18 \, x e^{\left (4 i \, a\right )}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 83, normalized size = 1.30 \[ -\frac {x^{5}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac {11 \, x^{3} e^{\left (2 i \, a\right )}}{3 \, {\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac {6 \, \arctan \left (\frac {x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right ) e^{\left (4 i \, a\right )}}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {10 \, x e^{\left (4 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 48, normalized size = 0.75 \[ -\frac {7 x^{3}}{3}-\frac {2 x^{3}}{\frac {{\mathrm e}^{2 i a}}{x^{2}}-1}-6 \,{\mathrm e}^{2 i a} x +6 \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 352, normalized size = 5.50 \[ -\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 \relax (a), x + \cos \relax (a)\right ) + {\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\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 \relax (a), x + \cos \relax (a)\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 \relax (a), x - \cos \relax (a)\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 \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{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 \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right )}{6 \, x^{2} - 6 \, \cos \left (2 \, a\right ) - 6 i \, \sin \left (2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 57, normalized size = 0.89 \[ -{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,6{}\mathrm {i}-\frac {x^3}{3}-4\,x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}-\frac {2\,x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}}{{\mathrm {e}}^{a\,2{}\mathrm {i}}-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 60, normalized size = 0.94 \[ - \frac {x^{3}}{3} - 4 x e^{2 i a} + \frac {2 x e^{4 i a}}{x^{2} - e^{2 i a}} - 3 \left (\log {\left (x - e^{i a} \right )} - \log {\left (x + e^{i a} \right )}\right ) e^{3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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