Optimal. Leaf size=45 \[ -\frac{2 i e^{-2 i a}}{x}+2 i e^{-3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i}{3 x^3} \]
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Rubi [F] time = 0.025074, 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 = {} \[ \int \frac{\cot (a+i \log (x))}{x^4} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cot (a+i \log (x))}{x^4} \, dx &=\int \frac{\cot (a+i \log (x))}{x^4} \, dx\\ \end{align*}
Mathematica [A] time = 0.0215919, size = 70, normalized size = 1.56 \[ -\frac{2 \sin (2 a)}{x}-\frac{2 i \cos (2 a)}{x}+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 59, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{3}}}{{x}^{3}}}+i \left ( -2\,{\frac{1}{ \left ({{\rm e}^{ia}} \right ) ^{2}x}}+{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{3}}}-{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{3}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14161, size = 192, normalized size = 4.27 \begin{align*} -\frac{3 \, x^{3}{\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + 3 \, x^{3}{\left (i \, \cos \left (3 \, a\right ) + \sin \left (3 \, 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 (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{3} + 12 \, x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} + 2 i}{6 \, x^{3}} \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^{4} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.626059, size = 54, normalized size = 1.2 \begin{align*} - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{- 3 i a} - \frac{\left (6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37743, size = 66, normalized size = 1.47 \begin{align*} i \, e^{\left (-3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac{2 i \, e^{\left (-2 i \, a\right )}}{x} - \frac{i}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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