Optimal. Leaf size=53 \[ \frac {\sqrt {2+3 x^2} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {2+3 x^2}{1+x^2}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {429}
\begin {gather*} \frac {\sqrt {3 x^2+2} F\left (\text {ArcTan}(x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {-x^2-1} \sqrt {\frac {3 x^2+2}{x^2+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 429
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+3 x^2}} \, dx &=\frac {\sqrt {2+3 x^2} F\left (\tan ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {2+3 x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 39, normalized size = 0.74 \begin {gather*} -\frac {i \sqrt {1+x^2} F\left (i \sinh ^{-1}(x)|\frac {3}{2}\right )}{\sqrt {2} \sqrt {-1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 36, normalized size = 0.68
method | result | size |
default | \(\frac {i \EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {\sqrt {6}}{3}\right ) \sqrt {3}\, \sqrt {-x^{2}-1}}{3 \sqrt {x^{2}+1}}\) | \(36\) |
elliptic | \(-\frac {i \sqrt {-\left (3 x^{2}+2\right ) \left (x^{2}+1\right )}\, \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {\sqrt {6}}{3}\right )}{6 \sqrt {-x^{2}-1}\, \sqrt {3 x^{2}+2}\, \sqrt {-3 x^{4}-5 x^{2}-2}}\) | \(84\) |
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 = 0.20, size = 10, normalized size = 0.19 \begin {gather*} \frac {1}{2} i \, \sqrt {-2} {\rm ellipticF}\left (i \, x, \frac {3}{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^{2} - 1} \sqrt {3 x^{2} + 2}}\, 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 {-x^2-1}\,\sqrt {3\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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