Optimal. Leaf size=40 \[ \frac {2 i \log (-a x+i)}{a^3}+\frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3} \]
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Rubi [A] time = 0.03, antiderivative size = 40, 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} \[ \frac {2 x}{a^2}+\frac {2 i \log (-a x+i)}{a^3}-\frac {i x^2}{a}-\frac {x^3}{3} \]
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^2 \, dx &=\int \frac {x^2 (1-i a x)}{1+i a x} \, dx\\ &=\int \left (\frac {2}{a^2}-\frac {2 i x}{a}-x^2+\frac {2 i}{a^2 (-i+a x)}\right ) \, dx\\ &=\frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3}+\frac {2 i \log (i-a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {2 i \log (-a x+i)}{a^3}+\frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 37, normalized size = 0.92 \[ -\frac {a^{3} x^{3} + 3 i \, a^{2} x^{2} - 6 \, a x - 6 i \, \log \left (\frac {a x - i}{a}\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 68, normalized size = 1.70 \[ -\frac {2 \, i \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a^{3}} - \frac {{\left (a i x + 1\right )}^{3} {\left (\frac {6 \, i^{2}}{a i x + 1} - \frac {15 \, i^{2}}{{\left (a i x + 1\right )}^{2}} + 1\right )}}{3 \, a^{3} i^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 1.18 \[ -\frac {x^{3}}{3}-\frac {i x^{2}}{a}+\frac {2 x}{a^{2}}+\frac {i \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {2 \arctan \left (a x \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 35, normalized size = 0.88 \[ -\frac {a^{2} x^{3} + 3 i \, a x^{2} - 6 \, x}{3 \, a^{2}} + \frac {2 i \, \log \left (i \, a x + 1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 36, normalized size = 0.90 \[ \frac {\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a^3}+\frac {2\,x}{a^2}-\frac {x^3}{3}-\frac {x^2\,1{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 32, normalized size = 0.80 \[ - \frac {x^{3}}{3} - \frac {i x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 i \log {\left (i a x + 1 \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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