Optimal. Leaf size=36 \[ \frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-x^2-1}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {421, 419} \[ \frac {\sqrt {x^2+1} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )}{2 \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx &=\frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}}\\ &=\frac {\sqrt {1+x^2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )}{2 \sqrt {-1-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 1.00 \[ \frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}}{2 \, {\left (2 \, x^{4} + x^{2} - 1\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 0.94 \[ \frac {i \sqrt {2}\, \sqrt {-x^{2}-1}\, \EllipticF \left (i x , i \sqrt {2}\right )}{2 \sqrt {x^{2}+1}} \]
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 \[ \int \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-4\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.78, size = 44, normalized size = 1.22 \[ \frac {\sqrt {2} \left (\begin {cases} - \frac {\sqrt {2} i F\left (\operatorname {asin}{\left (\sqrt {2} x \right )}\middle | - \frac {1}{2}\right )}{2} & \text {for}\: x > - \frac {\sqrt {2}}{2} \wedge x < \frac {\sqrt {2}}{2} \end {cases}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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