Optimal. Leaf size=25 \[ \frac {1}{2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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 = {253, 222} \[ \frac {1}{2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 222
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x^2} \sqrt {2+2 x^2}} \, dx &=\frac {\sqrt {-2+2 x^4} \int \frac {1}{\sqrt {-2+2 x^4}} \, dx}{\sqrt {-1+x^2} \sqrt {2+2 x^2}}\\ &=\frac {1}{2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 1.84 \[ \frac {x \sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )}{\sqrt {x^2-1} \sqrt {2 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{2} + 2} \sqrt {x^{2} - 1}}{2 \, {\left (x^{4} - 1\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{2} + 2} \sqrt {x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 30, normalized size = 1.20 \[ -\frac {i \sqrt {-x^{2}+1}\, \sqrt {2}\, \EllipticF \left (i x , i\right )}{2 \sqrt {x^{2}-1}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{2} + 2} \sqrt {x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.55, size = 75, normalized size = 3.00 \[ \frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} - \frac {\sqrt {2} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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