Optimal. Leaf size=29 \[ -\frac {\sqrt {x^2-1} \tanh ^{-1}(x)}{\sqrt {2} \sqrt {1-x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {23, 207} \[ -\frac {\sqrt {x^2-1} \tanh ^{-1}(x)}{\sqrt {2} \sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1+x^2}} \, dx &=\frac {\sqrt {-1+x^2} \int \frac {1}{-1+x^2} \, dx}{\sqrt {2-2 x^2}}\\ &=-\frac {\sqrt {-1+x^2} \tanh ^{-1}(x)}{\sqrt {2} \sqrt {1-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.38 \[ \frac {\left (x^2-1\right ) (\log (1-x)-\log (x+1))}{2 \sqrt {2} \sqrt {-\left (x^2-1\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 34, normalized size = 1.17 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x^{2} - 1} \sqrt {-2 \, x^{2} + 2} x}{x^{4} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.83 \[ \frac {\sqrt {2}\, \sqrt {-x^{2}+1}\, \arctanh \relax (x )}{2 \sqrt {x^{2}-1}} \]
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 \[ \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2-2\,x^2}} \,d x \]
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 \[ \frac {\sqrt {2} \int \frac {1}{\sqrt {1 - x^{2}} \sqrt {x^{2} - 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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