Optimal. Leaf size=65 \[ \frac {4 i}{a^4 (a x+i)}+\frac {16 \log (a x+i)}{a^4}+\frac {12 i x}{a^3}-\frac {4 x^2}{a^2}-\frac {4 i x^3}{3 a}+\frac {x^4}{4} \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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, 88} \[ -\frac {4 x^2}{a^2}+\frac {12 i x}{a^3}+\frac {4 i}{a^4 (a x+i)}+\frac {16 \log (a x+i)}{a^4}-\frac {4 i x^3}{3 a}+\frac {x^4}{4} \]
Antiderivative was successfully verified.
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Rule 88
Rule 5062
Rubi steps
\begin {align*} \int e^{4 i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (\frac {12 i}{a^3}-\frac {8 x}{a^2}-\frac {4 i x^2}{a}+x^3-\frac {4 i}{a^3 (i+a x)^2}+\frac {16}{a^3 (i+a x)}\right ) \, dx\\ &=\frac {12 i x}{a^3}-\frac {4 x^2}{a^2}-\frac {4 i x^3}{3 a}+\frac {x^4}{4}+\frac {4 i}{a^4 (i+a x)}+\frac {16 \log (i+a x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 65, normalized size = 1.00 \[ \frac {4 i}{a^4 (a x+i)}+\frac {16 \log (a x+i)}{a^4}+\frac {12 i x}{a^3}-\frac {4 x^2}{a^2}-\frac {4 i x^3}{3 a}+\frac {x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 70, normalized size = 1.08 \[ \frac {3 \, a^{5} x^{5} - 13 i \, a^{4} x^{4} - 32 \, a^{3} x^{3} + 96 i \, a^{2} x^{2} - 144 \, a x + {\left (192 \, a x + 192 i\right )} \log \left (\frac {a x + i}{a}\right ) + 48 i}{12 \, a^{5} x + 12 i \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 63, normalized size = 0.97 \[ \frac {16 \, \log \left (a x + i\right )}{a^{4}} + \frac {4 \, i}{{\left (a x + i\right )} a^{4}} + \frac {3 \, a^{8} x^{4} - 16 \, a^{7} i x^{3} - 48 \, a^{6} x^{2} + 144 \, a^{5} i x}{12 \, a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 70, normalized size = 1.08 \[ \frac {x^{4}}{4}-\frac {4 i x^{3}}{3 a}-\frac {4 x^{2}}{a^{2}}+\frac {12 i x}{a^{3}}+\frac {4 i}{a^{4} \left (a x +i\right )}+\frac {8 \ln \left (a^{2} x^{2}+1\right )}{a^{4}}-\frac {16 i \arctan \left (a x \right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 77, normalized size = 1.18 \[ -\frac {4 \, {\left (-i \, a x - 1\right )}}{a^{6} x^{2} + a^{4}} + \frac {3 \, a^{3} x^{4} - 16 i \, a^{2} x^{3} - 48 \, a x^{2} + 144 i \, x}{12 \, a^{3}} - \frac {16 i \, \arctan \left (a x\right )}{a^{4}} + \frac {8 \, \log \left (a^{2} x^{2} + 1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 60, normalized size = 0.92 \[ \frac {16\,\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )}{a^4}+\frac {x^4}{4}-\frac {4\,x^2}{a^2}+\frac {4{}\mathrm {i}}{a^5\,\left (x+\frac {1{}\mathrm {i}}{a}\right )}+\frac {x\,12{}\mathrm {i}}{a^3}-\frac {x^3\,4{}\mathrm {i}}{3\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 56, normalized size = 0.86 \[ \frac {x^{4}}{4} + \frac {4 i}{a^{5} x + i a^{4}} - \frac {4 i x^{3}}{3 a} - \frac {4 x^{2}}{a^{2}} + \frac {12 i x}{a^{3}} + \frac {16 \log {\left (a x + i \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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