Optimal. Leaf size=31 \[ \frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right ),-2\right )}{\sqrt {-x^2-1}} \]
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Rubi [A] time = 0.02, antiderivative size = 31, 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 = {421, 419} \[ \frac {\sqrt {x^2+1} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx &=\frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}}\\ &=\frac {\sqrt {1+x^2} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-1-x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 39, normalized size = 1.26 \[ -\frac {i \sqrt {x^2+1} \operatorname {EllipticF}\left (i \sinh ^{-1}(x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 2} \sqrt {-x^{2} - 1}}{x^{4} - x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 1.10 \[ \frac {i \sqrt {2}\, \sqrt {-x^{2}-1}\, \EllipticF \left (i x , \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {x^{2}+1}} \]
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 \[ \int \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 - x^{2}} \sqrt {- x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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