Optimal. Leaf size=27 \[ -\frac {1}{2} \sqrt {1-x^2} (2-x)-\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {785, 780, 216} \[ -\frac {1}{2} \sqrt {1-x^2} (2-x)-\frac {1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 785
Rubi steps
\begin {align*} \int \frac {x \sqrt {1-x^2}}{1+x} \, dx &=\int \frac {(1-x) x}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.96 \[ \left (\frac {x}{2}-1\right ) \sqrt {1-x^2}-\frac {1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 31, normalized size = 1.15 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} + \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 19, normalized size = 0.70 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} - \frac {1}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 1.26 \[ \frac {\sqrt {-x^{2}+1}\, x}{2}-\frac {\arcsin \relax (x )}{2}-\sqrt {2 x -\left (x +1\right )^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 28, normalized size = 1.04 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1} - \frac {1}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.74 \[ \left (\frac {x}{2}-1\right )\,\sqrt {1-x^2}-\frac {\mathrm {asin}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.35, size = 29, normalized size = 1.07 \[ \begin {cases} \frac {x \sqrt {1 - x^{2}}}{2} - \sqrt {1 - x^{2}} - \frac {\operatorname {asin}{\relax (x )}}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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