Optimal. Leaf size=64 \[ -\frac {3 x}{-x^2+e^{2 i a}}+\frac {e^{2 i a}}{x \left (-x^2+e^{2 i a}\right )}-2 e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right ) \]
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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 = {} \[ \int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx &=\int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.12, size = 72, normalized size = 1.12 \[ \frac {2 x (\cos (a)-i \sin (a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-2 \cos (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+2 i \sin (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 74, normalized size = 1.16 \[ -\frac {{\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - {\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - 3 \, x^{2} + e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 87, normalized size = 1.36 \[ 2 \, {\left (\frac {\arctan \left (\frac {x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right ) e^{\left (-2 i \, a\right )}}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {x e^{\left (-2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac {5 \, x^{2}}{x^{3} - x e^{\left (2 i \, a\right )}} - \frac {e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 38, normalized size = 0.59 \[ \frac {1}{x}-\frac {2}{x \left (\frac {{\mathrm e}^{2 i a}}{x^{2}}-1\right )}-2 \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 285, normalized size = 4.45 \[ -\frac {2 \, {\left ({\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x^{3} + {\left ({\left (2 \, {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (2 \, {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x - 6 \, x^{2} + {\left (x^{3} {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} - {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - {\left (x^{3} {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} - {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \cos \left (2 \, a\right ) - {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )}{2 \, x^{3} - x {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 47, normalized size = 0.73 \[ -\frac {2\,\mathrm {atanh}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}-3\,x^2}{x^3-x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 46, normalized size = 0.72 \[ - \frac {- 3 x^{2} + e^{2 i a}}{x^{3} - x e^{2 i a}} - \left (- \log {\left (x - e^{i a} \right )} + \log {\left (x + e^{i a} \right )}\right ) e^{- i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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