Optimal. Leaf size=29 \[ 2 i e^{-i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac {i}{x} \]
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Rubi [F] time = 0.03, 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 = {} \[ \int \frac {\tan (a+i \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan (a+i \log (x))}{x^2} \, dx &=\int \frac {\tan (a+i \log (x))}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.52 \[ 2 i \cos (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac {i}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 39, normalized size = 1.34 \[ -\frac {{\left (x \log \left (x + i \, e^{\left (i \, a\right )}\right ) - x \log \left (x - i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )}\right )} e^{\left (-i \, a\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 28, normalized size = 0.97 \[ -\frac {2 \, \arctan \left (\frac {i \, x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right )}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 0.83 \[ \frac {i}{x}+2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 127, normalized size = 4.38 \[ \frac {2 \, x {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + x {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 2 i}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.27, size = 27, normalized size = 0.93 \[ \frac {\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}+\frac {1{}\mathrm {i}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 27, normalized size = 0.93 \[ \left (\log {\left (x - i e^{i a} \right )} - \log {\left (x + i e^{i a} \right )}\right ) e^{- i a} + \frac {i}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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