Optimal. Leaf size=22 \[ -\frac{(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
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Rubi [A] time = 0.0412641, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5077} \[ -\frac{(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5077
Rubi steps
\begin{align*} \int \frac{e^{\tan ^{-1}(x)} x}{\left (1+x^2\right )^{3/2}} \, dx &=-\frac{e^{\tan ^{-1}(x)} (1-x)}{2 \sqrt{1+x^2}}\\ \end{align*}
Mathematica [C] time = 0.0099006, size = 37, normalized size = 1.68 \[ \frac{1}{2} (1-i x)^{-\frac{1}{2}+\frac{i}{2}} (1+i x)^{-\frac{1}{2}-\frac{i}{2}} (x-1) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.003, size = 16, normalized size = 0.7 \begin{align*}{\frac{ \left ( -1+x \right ){{\rm e}^{\arctan \left ( x \right ) }}}{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.664137, size = 53, normalized size = 2.41 \begin{align*} \frac{{\left (x - 1\right )} e^{\arctan \left (x\right )}}{2 \, \sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 119.363, size = 31, normalized size = 1.41 \begin{align*} \frac{x e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} - \frac{e^{\operatorname{atan}{\left (x \right )}}}{2 \sqrt{x^{2} + 1}} \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*} \int \frac{x e^{\arctan \left (x\right )}}{{\left (x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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