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 = 0.0246961, 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 = {} \[ \int \frac{\cot (a+i \log (x))}{x^2} \, dx \]
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*}
Mathematica [A] time = 0.0226852, 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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Maple [A] time = 0.06, size = 47, normalized size = 1.6 \begin{align*}{\frac{-i}{x}}+i \left ({\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{{{\rm e}^{ia}}}}-{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{{{\rm e}^{ia}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04772, size = 139, normalized size = 4.79 \begin{align*} \frac{x{\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + x{\left (-i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) -{\left ({\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (2 \, \cos \left (a\right ) - 2 i \, \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x - 2 i}{2 \, 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}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.474434, size = 29, normalized size = 1. \begin{align*} - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{- i a} - \frac{i}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26395, size = 54, normalized size = 1.86 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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