Optimal. Leaf size=20 \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
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Rubi [A] time = 0.0230173, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {5069} \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5069
Rubi steps
\begin{align*} \int \frac{e^{\tan ^{-1}(x)}}{\left (1+x^2\right )^{3/2}} \, dx &=\frac{e^{\tan ^{-1}(x)} (1+x)}{2 \sqrt{1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0054082, size = 20, normalized size = 1. \[ \frac{(x+1) e^{\tan ^{-1}(x)}}{2 \sqrt{x^2+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( x \right ) }} \left ( 1+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{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 = 0.930216, size = 53, normalized size = 2.65 \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 = 117.73, size = 31, normalized size = 1.55 \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{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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