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 = 0.0352676, antiderivative size = 41, normalized size of antiderivative = 1., 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 5073
Rule 47
Rule 41
Rule 215
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*}
Mathematica [A] time = 0.0577804, 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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Maple [B] time = 0.068, size = 143, normalized size = 3.5 \begin{align*}{\frac{-i}{{a}^{3}} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-2}}+{\frac{i}{a}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }}-{\ln \left ({ \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46689, size = 45, normalized size = 1.1 \begin{align*} -\frac{\operatorname{arsinh}\left (a x\right )}{a} + \frac{2 i \, \sqrt{a^{2} x^{2} + 1}}{i \, a^{2} x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95559, size = 126, normalized size = 3.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} x^{2} + 1}}{\left (i a x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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