Optimal. Leaf size=30 \[ -\frac {2 i x}{a}-\frac {x^2}{2}+\frac {2 \log (i-a x)}{a^2} \]
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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 = {5170, 78}
\begin {gather*} \frac {2 \log (-a x+i)}{a^2}-\frac {2 i x}{a}-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5170
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 \begin {gather*} -\frac {2 i x}{a}-\frac {x^2}{2}+\frac {2 \log (i-a x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 31, normalized size = 1.03
method | result | size |
default | \(-\frac {\frac {1}{2} a \,x^{2}+2 i x}{a}+\frac {2 \ln \left (-a x +i\right )}{a^{2}}\) | \(31\) |
risch | \(-\frac {x^{2}}{2}-\frac {2 i x}{a}+\frac {\ln \left (a^{2} x^{2}+1\right )}{a^{2}}+\frac {2 i \arctan \left (a x \right )}{a^{2}}\) | \(38\) |
meijerg | \(\frac {-\frac {i a x \left (2 a^{2} x^{2}+6 i a x +12\right )}{4 \left (i a x +1\right )}+3 \ln \left (i a x +1\right )}{a^{2}}-\frac {-\frac {i a x}{i a x +1}+\ln \left (i a x +1\right )}{a^{2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 0.93 \begin {gather*} \frac {i \, {\left (i \, a x^{2} - 4 \, x\right )}}{2 \, a} + \frac {2 \, \log \left (i \, a x + 1\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.67, size = 29, normalized size = 0.97 \begin {gather*} -\frac {a^{2} x^{2} + 4 i \, a x - 4 \, \log \left (\frac {a x - i}{a}\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.73 \begin {gather*} - \frac {x^{2}}{2} - \frac {2 i x}{a} + \frac {2 \log {\left (a x - i \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 52 vs. \(2 (24) = 48\).
time = 0.41, size = 52, normalized size = 1.73 \begin {gather*} -\frac {i \, {\left (\frac {{\left (i \, a x + 1\right )}^{2} {\left (-\frac {6 i}{i \, a x + 1} + i\right )}}{a} - \frac {4 i \, \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a}\right )}}{2 \, a} \end {gather*}
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 \begin {gather*} \frac {2\,\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )}{a^2}-\frac {x^2}{2}-\frac {x\,2{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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