Optimal. Leaf size=28 \[ \frac {1}{2 a (i+a x)}+\frac {\text {ArcTan}(a x)}{2 a} \]
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Rubi [A]
time = 0.03, 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 = {5181, 46, 209}
\begin {gather*} \frac {\text {ArcTan}(a x)}{2 a}+\frac {1}{2 a (a x+i)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 209
Rule 5181
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 = 0.01, size = 21, normalized size = 0.75 \begin {gather*} \frac {\frac {1}{i+a x}+\text {ArcTan}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 38, normalized size = 1.36
method | result | size |
risch | \(\frac {1}{2 a \left (a x +i\right )}+\frac {\arctan \left (a x \right )}{2 a}\) | \(24\) |
default | \(\frac {2 a^{2} x -2 i a}{4 a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 a}\) | \(38\) |
meijerg | \(\frac {\frac {2 x \sqrt {a^{2}}}{2 a^{2} x^{2}+2}+\frac {\sqrt {a^{2}}\, \arctan \left (a x \right )}{a}}{2 \sqrt {a^{2}}}+\frac {i a \,x^{2}}{2 a^{2} x^{2}+2}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 28, normalized size = 1.00 \begin {gather*} \frac {a x - i}{2 \, {\left (a^{3} x^{2} + a\right )}} + \frac {\arctan \left (a x\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 49 vs. \(2 (22) = 44\).
time = 2.50, size = 49, normalized size = 1.75 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 32, normalized size = 1.14 \begin {gather*} i \left (- \frac {i}{2 a^{2} x + 2 i a} + \frac {- \frac {\log {\left (x - \frac {i}{a} \right )}}{4} + \frac {\log {\left (x + \frac {i}{a} \right )}}{4}}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 36, normalized size = 1.29 \begin {gather*} \frac {i \, \log \left (a x + i\right )}{4 \, a} - \frac {i \, \log \left (-i \, a x - 1\right )}{4 \, a} + \frac {1}{2 \, {\left (a x + i\right )} a} \end {gather*}
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 \begin {gather*} \frac {1}{2\,\left (x\,a^2+a\,1{}\mathrm {i}\right )}+\frac {\mathrm {atan}\left (a\,x\right )}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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