Optimal. Leaf size=15 \[ \frac {i \log (a x+i)}{a} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5073, 31} \[ \frac {i \log (a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 5073
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx &=\int \frac {1}{1-i a x} \, dx\\ &=\frac {i \log (i+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {i \log (a x+i)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 15, normalized size = 1.00 \[ \frac {i \, \log \left (\frac {a x + i}{a}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 13, normalized size = 0.87 \[ \frac {i \log \left (-a i x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 1.73 \[ \frac {i \ln \left (a^{2} x^{2}+1\right )}{2 a}+\frac {\arctan \left (a x \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 24, normalized size = 1.60 \[ \frac {\arctan \left (a x\right )}{a} + \frac {i \, \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 15, normalized size = 1.00 \[ \frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,1{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 10, normalized size = 0.67 \[ \frac {i \log {\left (i a x - 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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