Optimal. Leaf size=48 \[ -\frac {2 \log (a x+i)}{a^4}-\frac {2 i x}{a^3}+\frac {x^2}{a^2}+\frac {2 i x^3}{3 a}-\frac {x^4}{4} \]
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Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5062, 77} \[ \frac {x^2}{a^2}-\frac {2 i x}{a^3}-\frac {2 \log (a x+i)}{a^4}+\frac {2 i x^3}{3 a}-\frac {x^4}{4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5062
Rubi steps
\begin {align*} \int e^{2 i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+i a x)}{1-i a x} \, dx\\ &=\int \left (-\frac {2 i}{a^3}+\frac {2 x}{a^2}+\frac {2 i x^2}{a}-x^3-\frac {2}{a^3 (i+a x)}\right ) \, dx\\ &=-\frac {2 i x}{a^3}+\frac {x^2}{a^2}+\frac {2 i x^3}{3 a}-\frac {x^4}{4}-\frac {2 \log (i+a x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.00 \[ -\frac {2 \log (a x+i)}{a^4}-\frac {2 i x}{a^3}+\frac {x^2}{a^2}+\frac {2 i x^3}{3 a}-\frac {x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 46, normalized size = 0.96 \[ -\frac {3 \, a^{4} x^{4} - 8 i \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 24 i \, a x + 24 \, \log \left (\frac {a x + i}{a}\right )}{12 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 48, normalized size = 1.00 \[ -\frac {3 \, a^{4} x^{4} - 8 \, a^{3} i x^{3} - 12 \, a^{2} x^{2} + 24 \, a i x}{12 \, a^{4}} - \frac {2 \, \log \left (a x + i\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 1.15 \[ -\frac {x^{4}}{4}+\frac {2 i x^{3}}{3 a}+\frac {x^{2}}{a^{2}}-\frac {2 i x}{a^{3}}-\frac {\ln \left (a^{2} x^{2}+1\right )}{a^{4}}+\frac {2 i \arctan \left (a x \right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 56, normalized size = 1.17 \[ -\frac {3 \, a^{3} x^{4} - 8 i \, a^{2} x^{3} - 12 \, a x^{2} + 24 i \, x}{12 \, a^{3}} + \frac {2 i \, \arctan \left (a x\right )}{a^{4}} - \frac {\log \left (a^{2} x^{2} + 1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 43, normalized size = 0.90 \[ \frac {x^2}{a^2}-\frac {x^4}{4}-\frac {2\,\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )}{a^4}-\frac {x\,2{}\mathrm {i}}{a^3}+\frac {x^3\,2{}\mathrm {i}}{3\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 0.88 \[ - \frac {x^{4}}{4} + \frac {2 i x^{3}}{3 a} + \frac {x^{2}}{a^{2}} - \frac {2 i x}{a^{3}} - \frac {2 \log {\left (i a x - 1 \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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