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

Optimal. Leaf size=22 \[ \cosh ^{-1}(x)-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0290471, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \cosh ^{-1}(x)-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.60591, size = 17, normalized size = 0.77 \[ \operatorname{acosh}{\left (x \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

acosh(x) - sqrt(x - 1)*sqrt(x + 1)/x

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Mathematica [A]  time = 0.0210917, size = 38, normalized size = 1.73 \[ \log \left (x+\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 44, normalized size = 2. \[{\frac{1}{x}\sqrt{-1+x}\sqrt{1+x} \left ( \ln \left ( x+\sqrt{{x}^{2}-1} \right ) x-\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52359, size = 36, normalized size = 1.64 \[ -\frac{\sqrt{x^{2} - 1}}{x} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22512, size = 80, normalized size = 3.64 \[ -\frac{{\left (\sqrt{x + 1} \sqrt{x - 1} x - x^{2}\right )} \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - 1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228453, size = 54, normalized size = 2.45 \[ -\frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} - \frac{1}{2} \,{\rm ln}\left ({\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/((sqrt(x + 1) - sqrt(x - 1))^4 + 4) - 1/2*ln((sqrt(x + 1) - sqrt(x - 1))^4)