Optimal. Leaf size=32 \[ \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|\frac {5}{2}\right )}{\sqrt {2} \sqrt {-1+x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {432, 430}
\begin {gather*} \frac {\sqrt {1-x^2} F\left (\text {ArcSin}(x)\left |\frac {5}{2}\right .\right )}{\sqrt {2} \sqrt {x^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-5 x^2} \sqrt {-1+x^2}} \, dx &=\frac {\sqrt {1-x^2} \int \frac {1}{\sqrt {2-5 x^2} \sqrt {1-x^2}} \, dx}{\sqrt {-1+x^2}}\\ &=\frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|\frac {5}{2}\right )}{\sqrt {2} \sqrt {-1+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 40, normalized size = 1.25 \begin {gather*} \frac {\sqrt {1-x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {5}{2}} x\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 29, normalized size = 0.91
method | result | size |
default | \(\frac {\EllipticF \left (x , \frac {\sqrt {10}}{2}\right ) \sqrt {-x^{2}+1}\, \sqrt {2}}{2 \sqrt {x^{2}-1}}\) | \(29\) |
elliptic | \(\frac {\sqrt {-\left (5 x^{2}-2\right ) \left (x^{2}-1\right )}\, \sqrt {-x^{2}+1}\, \sqrt {-10 x^{2}+4}\, \EllipticF \left (x , \frac {\sqrt {10}}{2}\right )}{2 \sqrt {-5 x^{2}+2}\, \sqrt {x^{2}-1}\, \sqrt {-5 x^{4}+7 x^{2}-2}}\) | \(74\) |
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 1.56, size = 37, normalized size = 1.16 \begin {gather*} \begin {cases} - \frac {\sqrt {5} i F\left (\operatorname {asin}{\left (\frac {\sqrt {10} x}{2} \right )}\middle | \frac {2}{5}\right )}{5} & \text {for}\: x > - \frac {\sqrt {10}}{5} \wedge x < \frac {\sqrt {10}}{5} \end {cases} \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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2-5\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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