Optimal. Leaf size=53 \[ -\frac {4}{a^3 (a x+i)}+\frac {12 i \log (a x+i)}{a^3}-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3} \]
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Rubi [A] time = 0.04, antiderivative size = 53, 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} \[ -\frac {8 x}{a^2}-\frac {4}{a^3 (a x+i)}+\frac {12 i \log (a x+i)}{a^3}-\frac {2 i x^2}{a}+\frac {x^3}{3} \]
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^2 \, dx &=\int \frac {x^2 (1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (-\frac {8}{a^2}-\frac {4 i x}{a}+x^2+\frac {4}{a^2 (i+a x)^2}+\frac {12 i}{a^2 (i+a x)}\right ) \, dx\\ &=-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3}-\frac {4}{a^3 (i+a x)}+\frac {12 i \log (i+a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.00 \[ -\frac {4}{a^3 (a x+i)}+\frac {12 i \log (a x+i)}{a^3}-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 62, normalized size = 1.17 \[ \frac {a^{4} x^{4} - 5 i \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 24 i \, a x - 36 \, {\left (-i \, a x + 1\right )} \log \left (\frac {a x + i}{a}\right ) - 12}{3 \, a^{4} x + 3 i \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 53, normalized size = 1.00 \[ \frac {12 \, i \log \left (a x + i\right )}{a^{3}} - \frac {4}{{\left (a x + i\right )} a^{3}} + \frac {a^{6} x^{3} - 6 \, a^{5} i x^{2} - 24 \, a^{4} x}{3 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 1.13 \[ \frac {x^{3}}{3}-\frac {2 i x^{2}}{a}-\frac {8 x}{a^{2}}-\frac {4}{a^{3} \left (a x +i\right )}+\frac {6 i \ln \left (a^{2} x^{2}+1\right )}{a^{3}}+\frac {12 \arctan \left (a x \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 68, normalized size = 1.28 \[ -\frac {8 \, a x - 8 i}{2 \, {\left (a^{5} x^{2} + a^{3}\right )}} + \frac {a^{2} x^{3} - 6 i \, a x^{2} - 24 \, x}{3 \, a^{2}} + \frac {12 \, \arctan \left (a x\right )}{a^{3}} + \frac {6 i \, \log \left (a^{2} x^{2} + 1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.96 \[ \frac {x^3}{3}+\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,12{}\mathrm {i}}{a^3}-\frac {8\,x}{a^2}-\frac {4}{a^4\,\left (x+\frac {1{}\mathrm {i}}{a}\right )}-\frac {x^2\,2{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 44, normalized size = 0.83 \[ \frac {x^{3}}{3} - \frac {4}{a^{4} x + i a^{3}} - \frac {2 i x^{2}}{a} - \frac {8 x}{a^{2}} + \frac {12 i \log {\left (a x + i \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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