Optimal. Leaf size=4 \[ x+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0428694, antiderivative size = 4, normalized size of antiderivative = 1., 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 2156
Rule 8
Rule 216
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*}
Mathematica [A] time = 0.0588855, size = 4, normalized size = 1. \[ x+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 51, normalized size = 12.8 \begin{align*} x+{\frac{\ln \left ( x-1 \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\it Artanh} \left ( x \right ) -{\frac{1}{2}\sqrt{- \left ( x-1 \right ) ^{2}-2\,x+2}}+\arcsin \left ( x \right ) +{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2\,x+2}} \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 \frac{x^{2}}{x^{2} + \sqrt{-x^{2} + 1} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.43393, size = 51, normalized size = 12.75 \begin{align*} x - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{x^{2} + \sqrt{1 - x^{2}} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21582, size = 5, normalized size = 1.25 \begin{align*} x + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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