Optimal. Leaf size=16 \[ -\log \left (\sqrt {1-x^2}+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 31
Rule 2155
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*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.00 \[ -\log \left (\sqrt {1-x^2}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 21, normalized size = 1.31 \[ -\log \relax (x) + \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 14, normalized size = 0.88 \[ -\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 59, normalized size = 3.69 \[ -\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\ln \relax (x )-\frac {\sqrt {2 x -\left (x +1\right )^{2}+2}}{2}-\frac {\sqrt {-2 x -\left (x -1\right )^{2}+2}}{2}+\sqrt {-x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 14, normalized size = 0.88 \[ -\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 21, normalized size = 1.31 \[ \ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.47, size = 44, normalized size = 2.75 \[ \frac {\log {\left (2 \sqrt {1 - x^{2}} \right )}}{2} - \frac {\log {\left (2 \sqrt {1 - x^{2}} + 2 \right )}}{2} - \frac {\log {\left (x^{2} - \sqrt {1 - x^{2}} - 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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