Optimal. Leaf size=55 \[ \sqrt{1-x^2}-\sqrt{1-x^2} \log \left (\sqrt{1-x^2}+1\right )-\log \left (\sqrt{1-x^2}+1\right ) \]
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Rubi [A] time = 0.0533209, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {261, 2554, 1591, 190, 43} \[ \sqrt{1-x^2}-\sqrt{1-x^2} \log \left (\sqrt{1-x^2}+1\right )-\log \left (\sqrt{1-x^2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 2554
Rule 1591
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{x \log \left (1+\sqrt{1-x^2}\right )}{\sqrt{1-x^2}} \, dx &=-\sqrt{1-x^2} \log \left (1+\sqrt{1-x^2}\right )-\int \frac{x}{1+\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2} \log \left (1+\sqrt{1-x^2}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{x}} \, dx,x,1-x^2\right )\\ &=-\sqrt{1-x^2} \log \left (1+\sqrt{1-x^2}\right )+\operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,\sqrt{1-x^2}\right )\\ &=-\sqrt{1-x^2} \log \left (1+\sqrt{1-x^2}\right )+\operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,\sqrt{1-x^2}\right )\\ &=\sqrt{1-x^2}-\log \left (1+\sqrt{1-x^2}\right )-\sqrt{1-x^2} \log \left (1+\sqrt{1-x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0207859, size = 41, normalized size = 0.75 \[ \sqrt{1-x^2}-\left (\sqrt{1-x^2}+1\right ) \log \left (\sqrt{1-x^2}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 37, normalized size = 0.7 \begin{align*} -\ln \left ( 1+\sqrt{-{x}^{2}+1} \right ) \left ( 1+\sqrt{-{x}^{2}+1} \right ) +1+\sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94243, size = 49, normalized size = 0.89 \begin{align*} -{\left (\sqrt{-x^{2} + 1} + 1\right )} \log \left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} + 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07777, size = 86, normalized size = 1.56 \begin{align*} -{\left (\sqrt{-x^{2} + 1} + 1\right )} \log \left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.99007, size = 31, normalized size = 0.56 \begin{align*} \sqrt{1 - x^{2}} - \left (\sqrt{1 - x^{2}} + 1\right ) \log{\left (\sqrt{1 - x^{2}} + 1 \right )} + 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08964, size = 49, normalized size = 0.89 \begin{align*} -{\left (\sqrt{-x^{2} + 1} + 1\right )} \log \left (\sqrt{-x^{2} + 1} + 1\right ) + \sqrt{-x^{2} + 1} + 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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