Optimal. Leaf size=20 \[ -x-\frac {2 i \log (-a x+i)}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5061, 43} \[ -x-\frac {2 i \log (-a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5061
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a x)} \, dx &=\int \frac {1-i a x}{1+i a x} \, dx\\ &=\int \left (-1-\frac {2 i}{-i+a x}\right ) \, dx\\ &=-x-\frac {2 i \log (i-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.50 \[ -\frac {i \log \left (a^2 x^2+1\right )}{a}+\frac {2 \tan ^{-1}(a x)}{a}-x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 21, normalized size = 1.05 \[ -\frac {a x + 2 i \, \log \left (\frac {a x - i}{a}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 67, normalized size = 3.35 \[ a^{2} {\left (\frac {{\left (a i x + 1\right )} i}{a^{3}} + \frac {2 \, i \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a^{3}} - \frac {i}{{\left (a i x + 1\right )} a^{3}}\right )} + \frac {i}{{\left (a i x + 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 1.50 \[ -x -\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}+\frac {2 \arctan \left (a x \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 16, normalized size = 0.80 \[ -x - \frac {2 i \, \log \left (i \, a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 19, normalized size = 0.95 \[ -x-\frac {\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.75 \[ - x - \frac {2 i \log {\left (i a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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