Optimal. Leaf size=31 \[ \frac {4}{a (a x+i)}-\frac {4 i \log (a x+i)}{a}+x \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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} \[ \frac {4}{a (a x+i)}-\frac {4 i \log (a x+i)}{a}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 5061
Rubi steps
\begin {align*} \int e^{4 i \tan ^{-1}(a x)} \, dx &=\int \frac {(1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (1-\frac {4}{(i+a x)^2}-\frac {4 i}{i+a x}\right ) \, dx\\ &=x+\frac {4}{a (i+a x)}-\frac {4 i \log (i+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.35 \[ -\frac {2 i \log \left (a^2 x^2+1\right )}{a}+\frac {4}{a (a x+i)}-\frac {4 \tan ^{-1}(a x)}{a}+x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 43, normalized size = 1.39 \[ \frac {a^{2} x^{2} + i \, a x - 4 \, {\left (i \, a x - 1\right )} \log \left (\frac {a x + i}{a}\right ) + 4}{a^{2} x + i \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 26, normalized size = 0.84 \[ x - \frac {4 \, i \log \left (a x + i\right )}{a} + \frac {4}{{\left (a x + i\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 1.32 \[ x +\frac {4}{a \left (a x +i\right )}-\frac {2 i \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {4 \arctan \left (a x \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 45, normalized size = 1.45 \[ x + \frac {8 \, a x - 8 i}{2 \, {\left (a^{3} x^{2} + a\right )}} - \frac {4 \, \arctan \left (a x\right )}{a} - \frac {2 i \, \log \left (a^{2} x^{2} + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 32, normalized size = 1.03 \[ x+\frac {4}{a^2\,\left (x+\frac {1{}\mathrm {i}}{a}\right )}-\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,4{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 22, normalized size = 0.71 \[ x + \frac {4}{a^{2} x + i a} - \frac {4 i \log {\left (a x + i \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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