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

Optimal. Leaf size=36 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0563378, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.54127, size = 29, normalized size = 0.81 \[ \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0138223, size = 62, normalized size = 1.72 \[ -\frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} (x+1)+x \sqrt{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{x-1} \sqrt{x+1}}\right )\right )}{\sqrt{x-1} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 59, normalized size = 1.6 \[{\frac{1+x}{x}\sqrt{{\frac{-1+x}{1+x}}} \left ( \left ({x}^{2}-1 \right ) ^{{\frac{3}{2}}}-{x}^{2}\sqrt{{x}^{2}-1}-\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x \right ){\frac{1}{\sqrt{ \left ( -1+x \right ) \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((-1+x)/(1+x))^(1/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.797029, size = 55, normalized size = 1.53 \[ -\frac{2 \, \sqrt{\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} + 1} + 2 \, \arctan \left (\sqrt{\frac{x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284026, size = 49, normalized size = 1.36 \[ \frac{2 \, x \arctan \left (\sqrt{\frac{x - 1}{x + 1}}\right ) -{\left (x + 1\right )} \sqrt{\frac{x - 1}{x + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269447, size = 69, normalized size = 1.92 \[ -\frac{1}{2} \,{\left (\pi - 2\right )}{\rm sign}\left (x + 1\right ) + 2 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right ){\rm sign}\left (x + 1\right ) - \frac{2 \,{\rm sign}\left (x + 1\right )}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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