Optimal. Leaf size=4 \[ x+\sin ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2156, 8, 216} \[ x+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 216
Rule 2156
Rubi steps
\begin {align*} \int \frac {x^2}{-1+x^2+\sqrt {1-x^2}} \, dx &=-\int -1 \, dx+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=x+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 4, normalized size = 1.00 \[ x+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 20, normalized size = 5.00 \[ x - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 4, normalized size = 1.00 \[ x + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 51, normalized size = 12.75 \[ x +\arctanh \relax (x )+\arcsin \relax (x )+\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}-\frac {\sqrt {-2 x -\left (x -1\right )^{2}+2}}{2}+\frac {\sqrt {2 x -\left (x +1\right )^{2}+2}}{2} \]
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 \[ \int \frac {x^{2}}{x^{2} + \sqrt {-x^{2} + 1} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 4, normalized size = 1.00 \[ x+\mathrm {asin}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{x^{2} + \sqrt {1 - x^{2}} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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