Optimal. Leaf size=22 \[ -\frac {(1-x) e^{\tan ^{-1}(x)}}{2 \sqrt {x^2+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 0.43, size = 15, normalized size = 0.68 \[ \frac {{\left (x - 1\right )} e^{\arctan \relax (x)}}{2 \, \sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 24, normalized size = 1.09 \[ \frac {1}{2} \, {\left (\frac {x}{\sqrt {x^{2} + 1}} - \frac {1}{\sqrt {x^{2} + 1}}\right )} e^{\arctan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 0.73 \[ \frac {\left (x -1\right ) {\mathrm e}^{\arctan \relax (x )}}{2 \sqrt {x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{\arctan \relax (x)}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x\,{\mathrm {e}}^{\mathrm {atan}\relax (x)}}{{\left (x^2+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.84, size = 31, normalized size = 1.41 \[ \frac {x e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} - \frac {e^{\operatorname {atan}{\relax (x )}}}{2 \sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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