Optimal. Leaf size=16 \[ \frac {1}{2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {430}
\begin {gather*} \frac {1}{2} F\left (\text {ArcSin}\left (\sqrt {2} x\right )|-\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1+x^2}} \, dx &=\frac {1}{2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 16, normalized size = 1.00 \begin {gather*} \frac {1}{2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 15, normalized size = 0.94
method | result | size |
default | \(\frac {\EllipticF \left (x \sqrt {2}, \frac {i \sqrt {2}}{2}\right )}{2}\) | \(15\) |
elliptic | \(\frac {\sqrt {-\left (2 x^{2}-1\right ) \left (x^{2}+1\right )}\, \sqrt {2}\, \EllipticF \left (x \sqrt {2}, \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {-4 x^{4}-2 x^{2}+2}}\) | \(48\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.27, size = 9, normalized size = 0.56 \begin {gather*} \frac {1}{2} \, {\rm ellipticF}\left (\sqrt {2} x, -\frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.09, size = 41, normalized size = 2.56 \begin {gather*} \frac {\sqrt {2} \left (\begin {cases} \frac {\sqrt {2} 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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-4\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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