3.740 \(\int \frac{\sqrt{\frac{-2 x^2+x^4}{\left (-1+x^2\right )^2}}}{2+x^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{\left (1-x^2\right ) \sqrt{-\frac{2 x^2-x^4}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt{x^2-2}\right )}{3 x \sqrt{x^2-2}}-\frac{2 \left (1-x^2\right ) \sqrt{-\frac{2 x^2-x^4}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\frac{\sqrt{x^2-2}}{2}\right )}{3 x \sqrt{x^2-2}} \]

[Out]

(-2*(1 - x^2)*Sqrt[-((2*x^2 - x^4)/(1 - x^2)^2)]*ArcTan[Sqrt[-2 + x^2]/2])/(3*x*
Sqrt[-2 + x^2]) + ((1 - x^2)*Sqrt[-((2*x^2 - x^4)/(1 - x^2)^2)]*ArcTan[Sqrt[-2 +
x^2]])/(3*x*Sqrt[-2 + x^2])

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Rubi [A]  time = 0.517798, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\left (1-x^2\right ) \sqrt{-\frac{2 x^2-x^4}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\sqrt{x^2-2}\right )}{3 x \sqrt{x^2-2}}-\frac{2 \left (1-x^2\right ) \sqrt{-\frac{2 x^2-x^4}{\left (1-x^2\right )^2}} \tan ^{-1}\left (\frac{\sqrt{x^2-2}}{2}\right )}{3 x \sqrt{x^2-2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-2*x^2 + x^4)/(-1 + x^2)^2]/(2 + x^2),x]

[Out]

(-2*(1 - x^2)*Sqrt[-((2*x^2 - x^4)/(1 - x^2)^2)]*ArcTan[Sqrt[-2 + x^2]/2])/(3*x*
Sqrt[-2 + x^2]) + ((1 - x^2)*Sqrt[-((2*x^2 - x^4)/(1 - x^2)^2)]*ArcTan[Sqrt[-2 +
x^2]])/(3*x*Sqrt[-2 + x^2])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{2} i \int \frac{\sqrt{\frac{x^{4} - 2 x^{2}}{\left (x^{2} - 1\right )^{2}}}}{- 4 x + 4 \sqrt{2} i}\, dx + \frac{\sqrt{2} i \int \frac{\sqrt{\frac{x^{4} - 2 x^{2}}{\left (x^{2} - 1\right )^{2}}}}{x + \sqrt{2} i}\, dx}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((x**4-2*x**2)/(x**2-1)**2)**(1/2)/(x**2+2),x)

[Out]

sqrt(2)*I*Integral(sqrt((x**4 - 2*x**2)/(x**2 - 1)**2)/(-4*x + 4*sqrt(2)*I), x)
+ sqrt(2)*I*Integral(sqrt((x**4 - 2*x**2)/(x**2 - 1)**2)/(x + sqrt(2)*I), x)/4

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Mathematica [A]  time = 0.0402558, size = 70, normalized size = 0.57 \[ \frac{\sqrt{\frac{x^2 \left (x^2-2\right )}{\left (x^2-1\right )^2}} \left (x^2-1\right ) \left (2 \tan ^{-1}\left (\frac{\sqrt{x^2-2}}{2}\right )-\tan ^{-1}\left (\sqrt{x^2-2}\right )\right )}{3 x \sqrt{x^2-2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(-2*x^2 + x^4)/(-1 + x^2)^2]/(2 + x^2),x]

[Out]

(Sqrt[(x^2*(-2 + x^2))/(-1 + x^2)^2]*(-1 + x^2)*(2*ArcTan[Sqrt[-2 + x^2]/2] - Ar
cTan[Sqrt[-2 + x^2]]))/(3*x*Sqrt[-2 + x^2])

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Maple [A]  time = 0.012, size = 75, normalized size = 0.6 \[{\frac{{x}^{2}-1}{6\,x}\sqrt{{\frac{{x}^{2} \left ({x}^{2}-2 \right ) }{ \left ({x}^{2}-1 \right ) ^{2}}}} \left ( \arctan \left ({(2+x){\frac{1}{\sqrt{{x}^{2}-2}}}} \right ) -\arctan \left ({(x-2){\frac{1}{\sqrt{{x}^{2}-2}}}} \right ) +4\,\arctan \left ( 1/2\,\sqrt{{x}^{2}-2} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((x^4-2*x^2)/(x^2-1)^2)^(1/2)/(x^2+2),x)

[Out]

1/6*(x^2*(x^2-2)/(x^2-1)^2)^(1/2)*(x^2-1)*(arctan((2+x)/(x^2-2)^(1/2))-arctan((x
-2)/(x^2-2)^(1/2))+4*arctan(1/2*(x^2-2)^(1/2)))/x/(x^2-2)^(1/2)

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Maxima [A]  time = 0.815802, size = 31, normalized size = 0.25 \[ \frac{2}{3} \, \arctan \left (\frac{1}{2} \, \sqrt{x^{2} - 2}\right ) - \frac{1}{3} \, \arctan \left (\sqrt{x^{2} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x^4 - 2*x^2)/(x^2 - 1)^2)/(x^2 + 2),x, algorithm="maxima")

[Out]

2/3*arctan(1/2*sqrt(x^2 - 2)) - 1/3*arctan(sqrt(x^2 - 2))

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Fricas [A]  time = 0.279032, size = 221, normalized size = 1.8 \[ \frac{1}{3} \, \arctan \left (\frac{x^{3} -{\left (x^{3} - x\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}} - 2 \, x}{x^{2} -{\left (x^{2} - 1\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}}\right ) - \frac{2}{3} \, \arctan \left (\frac{x^{3} -{\left (x^{3} - x\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}} - 2 \, x}{2 \,{\left (x^{2} -{\left (x^{2} - 1\right )} \sqrt{\frac{x^{4} - 2 \, x^{2}}{x^{4} - 2 \, x^{2} + 1}}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x^4 - 2*x^2)/(x^2 - 1)^2)/(x^2 + 2),x, algorithm="fricas")

[Out]

1/3*arctan((x^3 - (x^3 - x)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1)) - 2*x)/(x^2 -
(x^2 - 1)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1)))) - 2/3*arctan(1/2*(x^3 - (x^3 -
x)*sqrt((x^4 - 2*x^2)/(x^4 - 2*x^2 + 1)) - 2*x)/(x^2 - (x^2 - 1)*sqrt((x^4 - 2*
x^2)/(x^4 - 2*x^2 + 1))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((x**4-2*x**2)/(x**2-1)**2)**(1/2)/(x**2+2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269075, size = 45, normalized size = 0.37 \[ \frac{1}{3} \,{\left (2 \, \arctan \left (\frac{1}{2} \, \sqrt{x^{2} - 2}\right ) - \arctan \left (\sqrt{x^{2} - 2}\right )\right )}{\rm sign}\left (x^{3} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x^4 - 2*x^2)/(x^2 - 1)^2)/(x^2 + 2),x, algorithm="giac")

[Out]

1/3*(2*arctan(1/2*sqrt(x^2 - 2)) - arctan(sqrt(x^2 - 2)))*sign(x^3 - x)