3.832 \(\int \frac{\sqrt{-1+x} \sqrt{1+x}}{x} \, dx\)

Optimal. Leaf size=34 \[ \sqrt{x-1} \sqrt{x+1}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

[Out]

Sqrt[-1 + x]*Sqrt[1 + x] - ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

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Rubi [A]  time = 0.0445167, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \sqrt{x-1} \sqrt{x+1}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x]*Sqrt[1 + x] - ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 4.27578, size = 27, normalized size = 0.79 \[ \sqrt{x - 1} \sqrt{x + 1} - \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0303289, size = 34, normalized size = 1. \[ \sqrt{x-1} \sqrt{x+1}-2 \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x]*Sqrt[1 + x] - 2*ArcTan[Sqrt[-1 + x]/Sqrt[1 + x]]

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Maple [A]  time = 0.009, size = 35, normalized size = 1. \[{1\sqrt{-1+x}\sqrt{1+x} \left ( \sqrt{{x}^{2}-1}+\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.498, size = 18, normalized size = 0.53 \[ \sqrt{x^{2} - 1} + \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 - 1) + arcsin(1/abs(x))

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Fricas [A]  time = 0.245283, size = 96, normalized size = 2.82 \[ -\frac{\sqrt{x + 1} \sqrt{x - 1} x - x^{2} + 2 \,{\left (\sqrt{x + 1} \sqrt{x - 1} - x\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) + 1}{\sqrt{x + 1} \sqrt{x - 1} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225229, size = 43, normalized size = 1.26 \[ \sqrt{x + 1} \sqrt{x - 1} + 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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