Optimal. Leaf size=51 \[ \frac {\sqrt {2 x^2+1} \operatorname {EllipticF}\left (\tan ^{-1}(x),-1\right )}{\sqrt {2} \sqrt {-x^2-1} \sqrt {\frac {2 x^2+1}{x^2+1}}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, 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 = {418} \[ \frac {\sqrt {2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt {2} \sqrt {-x^2-1} \sqrt {\frac {2 x^2+1}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+4 x^2}} \, dx &=\frac {\sqrt {1+2 x^2} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {1+2 x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 37, normalized size = 0.73 \[ -\frac {i \sqrt {x^2+1} \operatorname {EllipticF}\left (i \sinh ^{-1}(x),2\right )}{\sqrt {2} \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {4 \, x^{2} + 2} \sqrt {-x^{2} - 1}}{2 \, {\left (2 \, x^{4} + 3 \, x^{2} + 1\right )}}, x\right ) \]
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 = 33, normalized size = 0.65 \[ \frac {i \sqrt {-x^{2}-1}\, \EllipticF \left (i \sqrt {2}\, x , \frac {\sqrt {2}}{2}\right )}{2 \sqrt {x^{2}+1}} \]
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 = -0.02 \[ \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {4\,x^2+2}} \,d x \]
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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