Optimal. Leaf size=45 \[ -\frac {2 i e^{-2 i a}}{x}-2 i e^{-3 i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac {i}{3 x^3} \]
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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 = {} \[ \int \frac {\tan (a+i \log (x))}{x^4} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan (a+i \log (x))}{x^4} \, dx &=\int \frac {\tan (a+i \log (x))}{x^4} \, dx\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 1.56 \[ -\frac {2 \sin (2 a)}{x}-\frac {2 i \cos (2 a)}{x}-2 i \cos (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac {i}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 53, normalized size = 1.18 \[ \frac {{\left (3 \, x^{3} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - 3 \, x^{3} \log \left (x - i \, e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} e^{\left (i \, a\right )} + i \, e^{\left (3 i \, a\right )}\right )} e^{\left (-3 i \, a\right )}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 28, normalized size = 0.62 \[ -2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-3 i \, a\right )} - \frac {2 i \, e^{\left (-2 i \, a\right )}}{x} + \frac {i}{3 \, x^{3}} \]
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 \[ \frac {i}{3 x^{3}}-\frac {2 i {\mathrm e}^{-2 i a}}{x}-2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-3 i a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 157, normalized size = 3.49 \[ -\frac {6 \, x^{3} {\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + x^{3} {\left (3 \, \cos \left (3 \, a\right ) - 3 i \, \sin \left (3 \, a\right )\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 12 \, x^{2} {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - 2 i}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.30, size = 40, normalized size = 0.89 \[ -\frac {\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}}-\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,2{}\mathrm {i}-\frac {1}{3}{}\mathrm {i}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 53, normalized size = 1.18 \[ \left (- \log {\left (x - i e^{i a} \right )} + \log {\left (x + i e^{i a} \right )}\right ) e^{- 3 i a} + \frac {\left (- 6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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