Optimal. Leaf size=28 \[ \sqrt {1-x^2}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\log (x) \]
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Rubi [A] time = 0.32, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used = {2103, 6688, 14, 266, 50, 63, 206} \[ \sqrt {1-x^2}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 50
Rule 63
Rule 206
Rule 266
Rule 2103
Rule 6688
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}+\sqrt {1+x}}{-\sqrt {1-x}+\sqrt {1+x}} \, dx &=\frac {1}{2} \int \frac {\sqrt {1-x} \left (\sqrt {1-x}+\sqrt {1+x}\right )}{x} \, dx+\frac {1}{2} \int \frac {\sqrt {1+x} \left (\sqrt {1-x}+\sqrt {1+x}\right )}{x} \, dx\\ &=\frac {1}{2} \int \frac {1-x+\sqrt {1-x^2}}{x} \, dx+\frac {1}{2} \int \frac {1+x+\sqrt {1-x^2}}{x} \, dx\\ &=\frac {1}{2} \int \left (-1+\frac {1}{x}+\frac {\sqrt {1-x^2}}{x}\right ) \, dx+\frac {1}{2} \int \left (1+\frac {1}{x}+\frac {\sqrt {1-x^2}}{x}\right ) \, dx\\ &=\log (x)+2 \left (\frac {1}{2} \int \frac {\sqrt {1-x^2}}{x} \, dx\right )\\ &=\log (x)+2 \left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,x^2\right )\right )\\ &=\log (x)+2 \left (\frac {\sqrt {1-x^2}}{2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\right )\\ &=\log (x)+2 \left (\frac {\sqrt {1-x^2}}{2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\right )\\ &=2 \left (\frac {\sqrt {1-x^2}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^2}\right )\right )+\log (x)\\ \end {align*}
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Mathematica [A] time = 0.16, size = 48, normalized size = 1.71 \[ \sqrt {1-x^2}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\log (x)+2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )+\sin ^{-1}(x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 36, normalized size = 1.29 \[ \sqrt {x + 1} \sqrt {-x + 1} + \log \relax (x) + \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.71 \[ \ln \relax (x )+\frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}} \]
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 \[ \int \frac {\sqrt {x + 1} + \sqrt {-x + 1}}{\sqrt {x + 1} - \sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 93, normalized size = 3.32 \[ \ln \left (\frac {{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}-1\right )-\ln \left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )+\ln \relax (x)-\frac {8\,\left (x-2\,\sqrt {x+1}+2\right )\,\left (x+2\,\sqrt {1-x}-2\right )}{{\left (2\,\sqrt {x+1}+2\,\sqrt {1-x}-4\right )}^2} \]
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 \[ - \int \frac {\sqrt {1 - x}}{\sqrt {1 - x} - \sqrt {x + 1}}\, dx - \int \frac {\sqrt {x + 1}}{\sqrt {1 - x} - \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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