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 = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 43
Rule 190
Rule 261
Rule 1591
Rule 2554
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.45, size = 35, normalized size = 0.64 \[ -{\left (\sqrt {-x^{2} + 1} + 1\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 36, normalized size = 0.65 \[ -{\left (\sqrt {-x^{2} + 1} + 1\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \sqrt {-x^{2} + 1} + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.67 \[ -\left (1+\sqrt {-x^{2}+1}\right ) \ln \left (1+\sqrt {-x^{2}+1}\right )+1+\sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 36, normalized size = 0.65 \[ -{\left (\sqrt {-x^{2} + 1} + 1\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \sqrt {-x^{2} + 1} + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 27, normalized size = 0.49 \[ -\left (\ln \left (\sqrt {1-x^2}+1\right )-1\right )\,\left (\sqrt {1-x^2}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.99, size = 31, normalized size = 0.56 \[ \sqrt {1 - x^{2}} - \left (\sqrt {1 - x^{2}} + 1\right ) \log {\left (\sqrt {1 - x^{2}} + 1 \right )} + 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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