3.730 \(\int \frac{\sqrt{\frac{x^2}{-1+x^2}}}{1+x^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{\sqrt{-\frac{x^2}{1-x^2}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

[Out]

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

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Rubi [A]  time = 0.177504, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{-\frac{x^2}{1-x^2}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.4958, size = 44, normalized size = 0.85 \[ \frac{\sqrt{2} \sqrt{\frac{x^{2}}{x^{2} - 1}} \sqrt{x^{2} - 1} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{x^{2} - 1}}{2} \right )}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(x**2/(x**2 - 1))*sqrt(x**2 - 1)*atan(sqrt(2)*sqrt(x**2 - 1)/2)/(2*x
)

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Mathematica [A]  time = 0.0298416, size = 49, normalized size = 0.94 \[ \frac{\sqrt{\frac{x^2}{x^2-1}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 42, normalized size = 0.8 \[{\frac{\sqrt{2}}{2\,x}\sqrt{{\frac{{x}^{2}}{{x}^{2}-1}}}\sqrt{{x}^{2}-1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{{x}^{2}-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{x^{2}}{x^{2} - 1}}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^2/(x^2 - 1))/(x^2 + 1), x)

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Fricas [A]  time = 0.272557, size = 81, normalized size = 1.56 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} x^{2} - \sqrt{2}{\left (x^{2} - 1\right )} \sqrt{\frac{x^{2}}{x^{2} - 1}}}{2 \,{\left (x \sqrt{\frac{x^{2}}{x^{2} - 1}} - x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{x^{2}}{x^{2} - 1}}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.281761, size = 55, normalized size = 1.06 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x^{2} - 1}\right ){\rm sign}\left (x^{2} - 1\right ){\rm sign}\left (x\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} i\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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