Optimal. Leaf size=30 \[ \frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt {2} \sqrt {x^2-1}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, 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 = {421, 419} \[ \frac {\sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2} \sqrt {x^2-1}} \]
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+4 x^2}} \, dx &=\frac {\sqrt {1-x^2} \int \frac {1}{\sqrt {1-x^2} \sqrt {2+4 x^2}} \, dx}{\sqrt {-1+x^2}}\\ &=\frac {\sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2} \sqrt {-1+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 1.00 \[ \frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt {2} \sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, 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 = 0.04, size = 34, normalized size = 1.13 \[ -\frac {i \sqrt {-x^{2}+1}\, \EllipticF \left (i \sqrt {2}\, x , \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {x^{2}-1}} \]
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 \[ \int \frac {1}{\sqrt {4 \, x^{2} + 2} \sqrt {x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {x^2-1}\,\sqrt {4\,x^2+2}} \,d x \]
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 \[ \frac {\sqrt {2} \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {2 x^{2} + 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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