Optimal. Leaf size=10 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),-1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.00, antiderivative size = 10, 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 = {248, 221} \[ \frac {F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 248
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-2 x^2} \sqrt {1+x^2}} \, dx &=\int \frac {1}{\sqrt {2-2 x^4}} \, dx\\ &=\frac {F\left (\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),-1\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 1} \sqrt {-2 \, x^{2} + 2}}{2 \, {\left (x^{4} - 1\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 10, normalized size = 1.00 \[ \frac {\sqrt {2}\, \EllipticF \left (x , i\right )}{2} \]
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 = -0.10 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.26, size = 76, normalized size = 7.60 \[ \frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} - \frac {\sqrt {2} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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