Optimal. Leaf size=27 \[ 2 i e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-i x \]
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Rubi [F] time = 0.01, 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 = {} \[ \int \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \cot (a+i \log (x)) \, dx &=\int \cot (a+i \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.56 \[ 2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-i x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 49, normalized size = 1.81 \[ -\sqrt {-e^{\left (2 i \, a\right )}} \log \left (x + i \, \sqrt {-e^{\left (2 i \, a\right )}}\right ) + \sqrt {-e^{\left (2 i \, a\right )}} \log \left (x - i \, \sqrt {-e^{\left (2 i \, a\right )}}\right ) - i \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 38, normalized size = 1.41 \[ i \, e^{\left (i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.81 \[ -i x +2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 98, normalized size = 3.63 \[ -\frac {1}{2} \, {\left (2 \, \cos \relax (a) + 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{2} \, {\left (2 \, \cos \relax (a) + 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) - \frac {1}{2} \, {\left (-i \, \cos \relax (a) + \sin \relax (a)\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - \frac {1}{2} \, {\left (i \, \cos \relax (a) - \sin \relax (a)\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - i \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.18, size = 29, normalized size = 1.07 \[ -x\,1{}\mathrm {i}+\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 29, normalized size = 1.07 \[ - i x - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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