Optimal. Leaf size=40 \[ \frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+2 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, 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 = {438, 435}
\begin {gather*} \frac {\sqrt {1-2 x^2} E\left (\text {ArcSin}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {2 x^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 435
Rule 438
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {-1+2 x^2}} \, dx &=\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-x^2}}{\sqrt {1-2 x^2}} \, dx}{\sqrt {-1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 35, normalized size = 0.88 \begin {gather*} \frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {-2+4 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 32, normalized size = 0.80
method | result | size |
default | \(\frac {\left (\EllipticF \left (x , \sqrt {2}\right )+\EllipticE \left (x , \sqrt {2}\right )\right ) \sqrt {-2 x^{2}+1}}{2 \sqrt {2 x^{2}-1}}\) | \(32\) |
elliptic | \(\frac {\sqrt {-\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \left (\frac {\sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{\sqrt {-2 x^{4}+3 x^{2}-1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \left (\EllipticF \left (x , \sqrt {2}\right )-\EllipticE \left (x , \sqrt {2}\right )\right )}{2 \sqrt {-2 x^{4}+3 x^{2}-1}}\right )}{\sqrt {-x^{2}+1}\, \sqrt {2 x^{2}-1}}\) | \(122\) |
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 = 0.15, size = 23, normalized size = 0.58 \begin {gather*} \frac {\sqrt {2 \, x^{2} - 1} \sqrt {-x^{2} + 1}}{2 \, x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {2 x^{2} - 1}}\, dx \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 = -0.02 \begin {gather*} \int \frac {\sqrt {1-x^2}}{\sqrt {2\,x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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