Optimal. Leaf size=78 \[ \sqrt {1-x^2}-\sqrt {1-x^2} \log \left (\sqrt {1-x^2}+x\right )-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {1-x^2}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.27, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {261, 2554, 6742, 2107, 321, 206, 444, 50, 63, 207, 388} \[ \sqrt {1-x^2}-\sqrt {1-x^2} \log \left (\sqrt {1-x^2}+x\right )-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {1-x^2}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 207
Rule 261
Rule 321
Rule 388
Rule 444
Rule 2107
Rule 2554
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \log \left (x+\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx &=-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )-\int \frac {x-\sqrt {1-x^2}}{x+\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )-\int \left (\frac {x}{x+\sqrt {1-x^2}}-\frac {\sqrt {1-x^2}}{x+\sqrt {1-x^2}}\right ) \, dx\\ &=-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )-\int \frac {x}{x+\sqrt {1-x^2}} \, dx+\int \frac {\sqrt {1-x^2}}{x+\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )+\int \frac {x^2}{1-2 x^2} \, dx-\int \frac {x \sqrt {1-x^2}}{1-2 x^2} \, dx+\int \left (\frac {x \sqrt {1-x^2}}{-1+2 x^2}-\frac {1-x^2}{-1+2 x^2}\right ) \, dx\\ &=-\frac {x}{2}-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )+\frac {1}{2} \int \frac {1}{1-2 x^2} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{1-2 x} \, dx,x,x^2\right )+\int \frac {x \sqrt {1-x^2}}{-1+2 x^2} \, dx-\int \frac {1-x^2}{-1+2 x^2} \, dx\\ &=\frac {\sqrt {1-x^2}}{2}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{2 \sqrt {2}}-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(1-2 x) \sqrt {1-x}} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {1}{-1+2 x^2} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{-1+2 x} \, dx,x,x^2\right )\\ &=\sqrt {1-x^2}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (-1+2 x)} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=\sqrt {1-x^2}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {1-x^2}\right )}{2 \sqrt {2}}-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=\sqrt {1-x^2}+\frac {\tanh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {1-x^2}\right )}{\sqrt {2}}-\sqrt {1-x^2} \log \left (x+\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 119, normalized size = 1.53 \[ \frac {1}{4} \left (4 \sqrt {1-x^2}-\sqrt {2} \log \left (\sqrt {2-2 x^2}-\sqrt {2} x+2\right )-\sqrt {2} \log \left (\sqrt {2-2 x^2}+\sqrt {2} x+2\right )-4 \sqrt {1-x^2} \log \left (\sqrt {1-x^2}+x\right )+2 \sqrt {2} \log \left (2 x+\sqrt {2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 115, normalized size = 1.47 \[ -\sqrt {-x^{2} + 1} \log \left (x + \sqrt {-x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {6 \, x^{2} - 2 \, \sqrt {2} {\left (2 \, x^{2} - 3\right )} + 2 \, \sqrt {-x^{2} + 1} {\left (3 \, \sqrt {2} - 4\right )} - 9}{2 \, x^{2} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {2 \, x^{2} + 2 \, \sqrt {2} x + 1}{2 \, x^{2} - 1}\right ) + \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 122, normalized size = 1.56 \[ -\sqrt {-x^{2} + 1} \log \left (x + \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 6 \right |}}{{\left | 4 \, \sqrt {2} + \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 6 \right |}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left ({\left | x + \frac {1}{2} \, \sqrt {2} \right |}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | x - \frac {1}{2} \, \sqrt {2} \right |}\right ) + \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x \ln \left (x +\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (x^{2} - 1\right )} \log \left (x + \sqrt {x + 1} \sqrt {-x + 1}\right )}{\sqrt {x + 1} \sqrt {-x + 1}} - \int \frac {{\left (x^{2} - 1\right )} e^{\left (-\frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (-x + 1\right )\right )}}{x}\,{d x} - \int \frac {1}{x^{2} + \sqrt {x + 1} x \sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\ln \left (x+\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log {\left (x + \sqrt {1 - x^{2}} \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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