Optimal. Leaf size=42 \[ \frac {(2+i a x) \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \sinh ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5060, 780, 215} \[ \frac {(2+i a x) \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 5060
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*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 0.90 \[ \frac {(2+i a x) \sqrt {a^2 x^2+1}-i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 43, normalized size = 1.02 \[ \frac {\sqrt {a^{2} x^{2} + 1} {\left (i \, a x + 2\right )} + i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 54, normalized size = 1.29 \[ \frac {1}{2} \, \sqrt {a^{2} x^{2} + 1} {\left (\frac {i x}{a} + \frac {2}{a^{2}}\right )} + \frac {i \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 69, normalized size = 1.64 \[ \frac {i x \sqrt {a^{2} x^{2}+1}}{2 a}-\frac {i \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {a^{2}}}+\frac {\sqrt {a^{2} x^{2}+1}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 42, normalized size = 1.00 \[ \frac {i \, \sqrt {a^{2} x^{2} + 1} x}{2 \, a} - \frac {i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} + \frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 51, normalized size = 1.21 \[ \frac {\left (\frac {1}{\sqrt {a^2}}+\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )\,\sqrt {a^2\,x^2+1}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{2\,a}}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.12, size = 51, normalized size = 1.21 \[ \begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\\frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases} + \frac {i x \sqrt {a^{2} x^{2} + 1}}{2 a} - \frac {i \operatorname {asinh}{\left (a x \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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