Optimal. Leaf size=30 \[ \frac {2 \log (-a x+i)}{a^2}-\frac {2 i x}{a}-\frac {x^2}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5062, 77} \[ \frac {2 \log (-a x+i)}{a^2}-\frac {2 i x}{a}-\frac {x^2}{2} \]
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 \, dx &=\int \frac {x (1-i a x)}{1+i a x} \, dx\\ &=\int \left (-\frac {2 i}{a}-x+\frac {2}{a (-i+a x)}\right ) \, dx\\ &=-\frac {2 i x}{a}-\frac {x^2}{2}+\frac {2 \log (i-a x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {2 \log (-a x+i)}{a^2}-\frac {2 i x}{a}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 29, normalized size = 0.97 \[ -\frac {a^{2} x^{2} + 4 i \, a x - 4 \, \log \left (\frac {a x - i}{a}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 58, normalized size = 1.93 \[ \frac {i {\left (\frac {4 \, i \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a} + \frac {{\left (a i x + 1\right )}^{2} {\left (i - \frac {6 \, i}{a i x + 1}\right )}}{a i^{2}}\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 1.27 \[ -\frac {x^{2}}{2}-\frac {2 i x}{a}+\frac {\ln \left (a^{2} x^{2}+1\right )}{a^{2}}+\frac {2 i \arctan \left (a x \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 28, normalized size = 0.93 \[ \frac {i \, {\left (i \, a x^{2} - 4 \, x\right )}}{2 \, a} + \frac {2 \, \log \left (i \, a x + 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 27, normalized size = 0.90 \[ \frac {2\,\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )}{a^2}-\frac {x^2}{2}-\frac {x\,2{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 24, normalized size = 0.80 \[ - \frac {x^{2}}{2} - \frac {2 i x}{a} + \frac {2 \log {\left (i a x + 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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