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 = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 204
Rule 288
Rule 1960
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.42, size = 55, normalized size = 1.08 \[ \frac {1}{2} \, {\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - \arctan \left (\frac {{\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 18, normalized size = 0.35 \[ \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 52, normalized size = 1.02 \[ \frac {\sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, \left (x^{2}+1\right ) \left (\arcsin \left (x^{2}\right )+\sqrt {-x^{4}+1}\right )}{2 \sqrt {-\left (x^{2}-1\right ) \left (x^{2}+1\right )}} \]
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 \[ \int x \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 55, normalized size = 1.08 \[ -\mathrm {atan}\left (\sqrt {-\frac {x^2-1}{x^2+1}}\right )-\frac {\sqrt {-\frac {x^2-1}{x^2+1}}}{\frac {x^2-1}{x^2+1}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.54, size = 39, normalized size = 0.76 \[ \begin {cases} \frac {\sqrt {1 - x^{2}} \sqrt {x^{2} + 1}}{2} - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {1 - x^{2}}}{2} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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