Optimal. Leaf size=29 \[ \frac{\tan ^{-1}(a x)}{2 a}-\frac{1}{2 a (-a x+i)} \]
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Rubi [A] time = 0.0412167, antiderivative size = 29, normalized size of antiderivative = 1., 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 5073
Rule 44
Rule 203
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) (1+i a x)^2} \, 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*}
Mathematica [A] time = 0.0158269, size = 21, normalized size = 0.72 \[ \frac{\tan ^{-1}(a x)+\frac{1}{a x-i}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 25, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ( -ax+i \right ) }}+{\frac{\arctan \left ( ax \right ) }{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8327, size = 116, normalized size = 4. \begin{align*} \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 \,{\left (a^{2} x - i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.40411, size = 36, normalized size = 1.24 \begin{align*} \frac{a}{2 a^{3} x - 2 i a^{2}} + \frac{- \frac{i \log{\left (x - \frac{i}{a} \right )}}{4} + \frac{i \log{\left (x + \frac{i}{a} \right )}}{4}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1191, size = 59, normalized size = 2.03 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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