Optimal. Leaf size=27 \[ i x-2 i e^{i a} \tan ^{-1}\left (e^{-i a} x\right ) \]
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Rubi [F] time = 0.0074411, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \tan (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \tan (a+i \log (x)) \, dx &=\int \tan (a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.0071506, size = 42, normalized size = 1.56 \[ -2 i \cos (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tan ^{-1}(x \cos (a)-i x \sin (a))+i x \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \tan \left ( a+i\ln \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.70817, size = 165, normalized size = 6.11 \begin{align*}{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \arctan \left (\frac{2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac{x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) - \frac{1}{2} \,{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \log \left (\frac{x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + i \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.50162, size = 27, normalized size = 1. \begin{align*} i x + \left (- \log{\left (x - i e^{i a} \right )} + \log{\left (x + i e^{i a} \right )}\right ) e^{i a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14532, size = 41, normalized size = 1.52 \begin{align*} \frac{2 \, \arctan \left (\frac{i \, x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right ) e^{\left (2 i \, a\right )}}{\sqrt{-e^{\left (2 i \, a\right )}}} + i \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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