Optimal. Leaf size=28 \[ \frac {\tan ^{-1}(a x)}{2 a}+\frac {1}{2 a (a x+i)} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5073, 44, 203} \[ \frac {\tan ^{-1}(a x)}{2 a}+\frac {1}{2 a (a x+i)} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 5073
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(1-i a x)^2 (1+i a x)} \, dx\\ &=\int \left (-\frac {1}{2 (i+a x)^2}+\frac {1}{2 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{2 a (i+a x)}+\frac {1}{2} \int \frac {1}{1+a^2 x^2} \, dx\\ &=\frac {1}{2 a (i+a x)}+\frac {\tan ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 0.75 \[ \frac {\tan ^{-1}(a x)+\frac {1}{a x+i}}{2 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 49, normalized size = 1.75 \[ \frac {{\left (i \, a x - 1\right )} \log \left (\frac {a x + i}{a}\right ) + {\left (-i \, a x + 1\right )} \log \left (\frac {a x - i}{a}\right ) + 2}{4 \, a^{2} x + 4 i \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 41, normalized size = 1.46 \[ \frac {i \log \left (a x + i\right )}{4 \, a} + \frac {\log \left (-a i x - 1\right )}{4 \, a i} + \frac {1}{2 \, {\left (a x + i\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 1.36 \[ \frac {2 a^{2} x -2 i a}{4 a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 28, normalized size = 1.00 \[ \frac {a x - i}{2 \, {\left (a^{3} x^{2} + a\right )}} + \frac {\arctan \left (a x\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 23, normalized size = 0.82 \[ \frac {1}{2\,\left (x\,a^2+a\,1{}\mathrm {i}\right )}+\frac {\mathrm {atan}\left (a\,x\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 34, normalized size = 1.21 \[ \frac {1}{2 a^{2} x + 2 i a} - \frac {\frac {i \log {\left (x - \frac {i}{a} \right )}}{4} - \frac {i \log {\left (x + \frac {i}{a} \right )}}{4}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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