Optimal. Leaf size=41 \[ -\frac {\sinh ^{-1}(a x)}{a}+\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5073, 47, 41, 215} \[ -\frac {\sinh ^{-1}(a x)}{a}+\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 47
Rule 215
Rule 5073
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx &=\int \frac {\sqrt {1-i a x}}{(1+i a x)^{3/2}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\int \frac {1}{\sqrt {1-i a x} \sqrt {1+i a x}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 i \sqrt {1-i a x}}{a \sqrt {1+i a x}}-\frac {\sinh ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 1.37 \[ \frac {2 \left (\sqrt {a^2 x^2+1}+(-1-i a x) \sin ^{-1}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a (a x-i)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 54, normalized size = 1.32 \[ \frac {2 \, a x + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} - 2 i}{a^{2} x - i \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 143, normalized size = 3.49 \[ -\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{a^{3} \left (x -\frac {i}{a}\right )^{2}}+\frac {i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a}-\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 33, normalized size = 0.80 \[ -\frac {\operatorname {arsinh}\left (a x\right )}{a} + \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{2} x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 56, normalized size = 1.37 \[ -\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {2\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ - \int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{2} x^{2} - 2 i a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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