Optimal. Leaf size=29 \[ \frac {\sinh ^{-1}(a x)}{a}-\frac {i \sqrt {a^2 x^2+1}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5059, 641, 215} \[ \frac {\sinh ^{-1}(a x)}{a}-\frac {i \sqrt {a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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Rule 215
Rule 641
Rule 5059
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a x)} \, dx &=\int \frac {1-i a x}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {i \sqrt {1+a^2 x^2}}{a}+\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {i \sqrt {1+a^2 x^2}}{a}+\frac {\sinh ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.90 \[ \frac {\sinh ^{-1}(a x)-i \sqrt {a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 37, normalized size = 1.28 \[ \frac {-i \, \sqrt {a^{2} x^{2} + 1} - \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 42, normalized size = 1.45 \[ -\frac {\sqrt {a^{2} x^{2} + 1} i}{a} - \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 97, normalized size = 3.34 \[ -\frac {i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 25, normalized size = 0.86 \[ \frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 32, normalized size = 1.10 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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