Optimal. Leaf size=8 \[ \frac {\tanh ^{-1}(x)}{\sqrt {2}} \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {22, 206} \begin {gather*} \frac {\tanh ^{-1}(x)}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 22
Rule 206
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-2 x^2} \sqrt {1-x^2}} \, dx &=\frac {\int \frac {1}{1-x^2} \, dx}{\sqrt {2}}\\ &=\frac {\tanh ^{-1}(x)}{\sqrt {2}}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 26, normalized size = 3.25 \begin {gather*} -\frac {\frac {1}{2} \log (1-x)-\frac {1}{2} \log (x+1)}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2-2 x^2} \sqrt {1-x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.30, size = 68, normalized size = 8.50 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {x^{6} + 5 \, x^{4} - 2 \, \sqrt {2} {\left (x^{3} + x\right )} \sqrt {-x^{2} + 1} \sqrt {-2 \, x^{2} + 2} - 5 \, x^{2} - 1}{x^{6} - 3 \, x^{4} + 3 \, x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 19, normalized size = 2.38 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (x + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2}\, \arctanh \relax (x )}{2} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.12 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.32, size = 22, normalized size = 2.75 \begin {gather*} - \sqrt {2} \left (\begin {cases} - \frac {\operatorname {acoth}{\relax (x )}}{2} & \text {for}\: x^{2} > 1 \\- \frac {\operatorname {atanh}{\relax (x )}}{2} & \text {for}\: x^{2} < 1 \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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