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 = 0.0072593, 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 = {} \[ \int \cot (a+i \log (x)) \, dx \]
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*}
Mathematica [A] time = 0.0083278, 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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Maple [A] time = 0.071, size = 44, normalized size = 1.6 \begin{align*} ix+i \left ( -2\,x+{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}+x \right ) -{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}-x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0568, size = 132, normalized size = 4.89 \begin{align*} -\frac{1}{2} \,{\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 ) - \frac{1}{2} \,{\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 ) - \frac{1}{2} \,{\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 ) - \frac{1}{2} \,{\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 ) - 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 = 0.494968, size = 29, normalized size = 1.07 \begin{align*} - i x - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + 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 [B] time = 1.34179, size = 51, normalized size = 1.89 \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 ) - i \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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