Optimal. Leaf size=51 \[ \frac{1}{2} \sqrt{\frac{1-x^2}{x^2+1}} \left (x^2+1\right )-\tan ^{-1}\left (\sqrt{\frac{1-x^2}{x^2+1}}\right ) \]
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Rubi [A] time = 0.0233902, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1960, 288, 204} \[ \frac{1}{2} \sqrt{\frac{1-x^2}{x^2+1}} \left (x^2+1\right )-\tan ^{-1}\left (\sqrt{\frac{1-x^2}{x^2+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 1960
Rule 288
Rule 204
Rubi steps
\begin{align*} \int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt{\frac{1-x^2}{1+x^2}}\right )\right )\\ &=\frac{1}{2} \sqrt{\frac{1-x^2}{1+x^2}} \left (1+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{\frac{1-x^2}{1+x^2}}\right )\\ &=\frac{1}{2} \sqrt{\frac{1-x^2}{1+x^2}} \left (1+x^2\right )-\tan ^{-1}\left (\sqrt{\frac{1-x^2}{1+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0313739, size = 86, normalized size = 1.69 \[ \frac{\sqrt{\frac{1-x^2}{x^2+1}} \sqrt{x^2+1} \left (\sqrt{x^2+1} \left (x^2-1\right )+2 \sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{1-x^2}}{\sqrt{2}}\right )\right )}{2 \left (x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 52, normalized size = 1. \begin{align*}{\frac{{x}^{2}+1}{2}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}+1}}} \left ( \sqrt{-{x}^{4}+1}+\arcsin \left ({x}^{2} \right ) \right ){\frac{1}{\sqrt{- \left ({x}^{2}-1 \right ) \left ({x}^{2}+1 \right ) }}}} \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 x \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49172, size = 134, normalized size = 2.63 \begin{align*} \frac{1}{2} \,{\left (x^{2} + 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}} - \arctan \left (\frac{{\left (x^{2} + 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 36.6693, size = 39, normalized size = 0.76 \begin{align*} \begin{cases} \frac{\sqrt{1 - x^{2}} \sqrt{x^{2} + 1}}{2} - \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{1 - x^{2}}}{2} \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19571, size = 24, normalized size = 0.47 \begin{align*} \frac{1}{2} \, \sqrt{-x^{4} + 1} + \frac{1}{2} \, \arcsin \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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