Optimal. Leaf size=83 \[ \frac {2 \sqrt {x^4-2 x^2} \tan ^{-1}\left (\frac {\sqrt {x^2-2}}{2}\right )}{3 x \sqrt {x^2-2}}-\frac {\sqrt {x^4-2 x^2} \tan ^{-1}\left (\sqrt {x^2-2}\right )}{3 x \sqrt {x^2-2}} \]
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Rubi [A] time = 0.15, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2056, 571, 83, 63, 203} \[ \frac {2 \sqrt {x^4-2 x^2} \tan ^{-1}\left (\frac {\sqrt {x^2-2}}{2}\right )}{3 x \sqrt {x^2-2}}-\frac {\sqrt {x^4-2 x^2} \tan ^{-1}\left (\sqrt {x^2-2}\right )}{3 x \sqrt {x^2-2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 83
Rule 203
Rule 571
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt {-2 x^2+x^4}}{\left (-1+x^2\right ) \left (2+x^2\right )} \, dx &=\frac {\sqrt {-2 x^2+x^4} \int \frac {x \sqrt {-2+x^2}}{\left (-1+x^2\right ) \left (2+x^2\right )} \, dx}{x \sqrt {-2+x^2}}\\ &=\frac {\sqrt {-2 x^2+x^4} \operatorname {Subst}\left (\int \frac {\sqrt {-2+x}}{(-1+x) (2+x)} \, dx,x,x^2\right )}{2 x \sqrt {-2+x^2}}\\ &=-\frac {\sqrt {-2 x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+x} (-1+x)} \, dx,x,x^2\right )}{6 x \sqrt {-2+x^2}}+\frac {\left (2 \sqrt {-2 x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+x} (2+x)} \, dx,x,x^2\right )}{3 x \sqrt {-2+x^2}}\\ &=-\frac {\sqrt {-2 x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}+\frac {\left (4 \sqrt {-2 x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{4+x^2} \, dx,x,\sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}\\ &=\frac {2 \sqrt {-2 x^2+x^4} \tan ^{-1}\left (\frac {1}{2} \sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}-\frac {\sqrt {-2 x^2+x^4} \tan ^{-1}\left (\sqrt {-2+x^2}\right )}{3 x \sqrt {-2+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 52, normalized size = 0.63 \[ -\frac {x \sqrt {x^2-2} \left (2 \tan ^{-1}\left (\frac {2}{\sqrt {x^2-2}}\right )+\tan ^{-1}\left (\sqrt {x^2-2}\right )\right )}{3 \sqrt {x^2 \left (x^2-2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 38, normalized size = 0.46 \[ -\frac {1}{3} \, \arctan \left (\frac {\sqrt {x^{4} - 2 \, x^{2}}}{x}\right ) + \frac {2}{3} \, \arctan \left (\frac {\sqrt {x^{4} - 2 \, x^{2}}}{2 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.36, size = 46, normalized size = 0.55 \[ \frac {1}{3} \, {\left (\arctan \left (i \, \sqrt {2}\right ) - 2 \, \arctan \left (\frac {1}{2} i \, \sqrt {2}\right )\right )} \mathrm {sgn}\relax (x) + \frac {2}{3} \, \arctan \left (\frac {1}{2} \, \sqrt {x^{2} - 2}\right ) \mathrm {sgn}\relax (x) - \frac {1}{3} \, \arctan \left (\sqrt {x^{2} - 2}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 0.76 \[ -\frac {\sqrt {x^{4}-2 x^{2}}\, \left (\arctan \left (\frac {x -2}{\sqrt {x^{2}-2}}\right )-\arctan \left (\frac {x +2}{\sqrt {x^{2}-2}}\right )-4 \arctan \left (\frac {\sqrt {x^{2}-2}}{2}\right )\right )}{6 \sqrt {x^{2}-2}\, x} \]
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 \[ \int \frac {\sqrt {x^{4} - 2 \, x^{2}}}{{\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x^4-2\,x^2}}{\left (x^2-1\right )\,\left (x^2+2\right )} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (x^{2} - 2\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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