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 = 0.03, 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 = {} \[ \int \frac {\cot (a+i \log (x))}{x^4} \, dx \]
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.59, size = 55, normalized size = 1.22 \[ \frac {{\left (3 i \, x^{3} e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - 3 i \, x^{3} e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} - i \, e^{\left (2 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 49, normalized size = 1.09 \[ i \, e^{\left (-3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac {2 i \, e^{\left (-2 i \, a\right )}}{x} - \frac {i}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 0.78 \[ -\frac {i}{3 x^{3}}-\frac {2 i {\mathrm e}^{-2 i a}}{x}+2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 142, normalized size = 3.16 \[ -\frac {3 \, x^{3} {\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + 3 \, x^{3} {\left (i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + {\left ({\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x^{3} + 12 \, x^{2} {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} + 2 i}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 44, normalized size = 0.98 \[ \frac {\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{{\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}}-\frac {2{}\mathrm {i}\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,x^2+\frac {1}{3}{}\mathrm {i}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 54, normalized size = 1.20 \[ - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{- 3 i a} - \frac {\left (6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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