Optimal. Leaf size=52 \[ \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {533, 432, 430}
\begin {gather*} \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F(\text {ArcSin}(x)|2)}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rule 533
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx &=\frac {\sqrt {-1+x^2} \int \frac {1}{\sqrt {-1+x^2} \sqrt {-1+2 x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {-1+x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {1-x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(52)=104\).
time = 33.71, size = 107, normalized size = 2.06 \begin {gather*} -\frac {2 (-1+x)^{3/2} \sqrt {\frac {1+x}{1-x}} \sqrt {\frac {1-2 x^2}{(-1+x)^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2+\sqrt {2}+\frac {1}{-1+x}}}{2^{3/4}}\right )|4 \left (-4+3 \sqrt {2}\right )\right )}{\sqrt {3+2 \sqrt {2}} \sqrt {1+x} \sqrt {-1+2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 58, normalized size = 1.12
method | result | size |
default | \(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \sqrt {2 x^{2}-1}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{2 x^{4}-3 x^{2}+1}\) | \(58\) |
elliptic | \(\frac {\sqrt {\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{\sqrt {-1+x}\, \sqrt {1+x}\, \sqrt {2 x^{2}-1}\, \sqrt {2 x^{4}-3 x^{2}+1}}\) | \(73\) |
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 = 0.92, size = 3, normalized size = 0.06 \begin {gather*} {\rm ellipticF}\left (x, 2\right ) \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 {1}{\sqrt {x - 1} \sqrt {x + 1} \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 {1}{\sqrt {2\,x^2-1}\,\sqrt {x-1}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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