Optimal. Leaf size=10 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {419} \[ \frac {F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+4 x^2}} \, dx &=\frac {F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 58, normalized size = 5.80 \[ -\frac {i \sqrt {1-x^2} \sqrt {2 x^2+1} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-2 x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {4 \, x^{2} + 2} \sqrt {-x^{2} + 1}}{2 \, {\left (2 \, x^{4} - x^{2} - 1\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.40 \[ \frac {\sqrt {2}\, \EllipticF \left (x , i \sqrt {2}\right )}{2} \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {4\,x^2+2}} \,d x \]
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 \[ \frac {\sqrt {2} \int \frac {1}{\sqrt {1 - x^{2}} \sqrt {2 x^{2} + 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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