Optimal. Leaf size=16 \[ -\log \left (\sqrt{1-x^2}+1\right ) \]
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Rubi [A] time = 0.04629, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2155, 31} \[ -\log \left (\sqrt{1-x^2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2155
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{1-x^2+\sqrt{1-x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{1-x}-x} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{1-x^2}\right )\\ &=-\log \left (1+\sqrt{1-x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0302715, size = 16, normalized size = 1. \[ -\log \left (\sqrt{1-x^2}+1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 59, normalized size = 3.7 \begin{align*} -\ln \left ( x \right ) +\sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) -{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}}-{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+2-2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928118, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02024, size = 53, normalized size = 3.31 \begin{align*} -\log \left (x\right ) + \log \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 [B] time = 2.74743, size = 44, normalized size = 2.75 \begin{align*} \frac{\log{\left (2 \sqrt{1 - x^{2}} \right )}}{2} - \frac{\log{\left (2 \sqrt{1 - x^{2}} + 2 \right )}}{2} - \frac{\log{\left (x^{2} - \sqrt{1 - x^{2}} - 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05177, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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