Optimal. Leaf size=29 \[ 2 i e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac {i}{x} \]
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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 = {} \[ \int \frac {\cot (a+i \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cot (a+i \log (x))}{x^2} \, dx &=\int \frac {\cot (a+i \log (x))}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.52 \[ 2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac {i}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 36, normalized size = 1.24 \[ \frac {i \, x e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, x e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 40, normalized size = 1.38 \[ i \, e^{\left (-i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac {i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 0.83 \[ -\frac {i}{x}+2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 103, normalized size = 3.55 \[ \frac {x {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + x {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - {\left ({\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x - 2 i}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 31, normalized size = 1.07 \[ -\frac {\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}-\frac {1{}\mathrm {i}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 29, normalized size = 1.00 \[ - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{- i a} - \frac {i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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