Optimal. Leaf size=42 \[ \frac{\sqrt{a^2 x^2+1} (2-i a x)}{2 a^2}+\frac{i \sinh ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.0199573, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5060, 780, 215} \[ \frac{\sqrt{a^2 x^2+1} (2-i a x)}{2 a^2}+\frac{i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 5060
Rule 780
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1-i a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{(2-i a x) \sqrt{1+a^2 x^2}}{2 a^2}+\frac{i \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{2 a}\\ &=\frac{(2-i a x) \sqrt{1+a^2 x^2}}{2 a^2}+\frac{i \sinh ^{-1}(a x)}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0261351, size = 38, normalized size = 0.9 \[ \frac{\sqrt{a^2 x^2+1} (2-i a x)+i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 152, normalized size = 3.6 \begin{align*}{\frac{-{\frac{i}{2}}x}{a}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{{\frac{i}{2}}}{a}\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{a}^{2}}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }}+{\frac{i}{a}\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 = 1.56325, size = 57, normalized size = 1.36 \begin{align*} -\frac{i \, \sqrt{a^{2} x^{2} + 1} x}{2 \, a} + \frac{i \, \operatorname{arsinh}\left (a x\right )}{2 \, a^{2}} + \frac{\sqrt{a^{2} x^{2} + 1}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76266, size = 103, normalized size = 2.45 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1}{\left (-i \, a x + 2\right )} - i \, \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, a^{2}} \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{x \sqrt{a^{2} x^{2} + 1}}{i a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13438, size = 73, normalized size = 1.74 \begin{align*} -\frac{1}{2} \, \sqrt{a^{2} x^{2} + 1}{\left (\frac{i x}{a} - \frac{2}{a^{2}}\right )} - \frac{i \log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, a{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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