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

Optimal. Leaf size=2 \[ \cosh ^{-1}(x) \]

[Out]

ArcCosh[x]

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Rubi [A]  time = 0.00834964, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcCosh[x]

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Rubi in Sympy [A]  time = 1.12774, size = 2, normalized size = 1. \[ \operatorname{acosh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

acosh(x)

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Mathematica [B]  time = 0.00704955, size = 16, normalized size = 8. \[ 2 \sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*ArcSinh[Sqrt[-1 + x]/Sqrt[2]]

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Maple [B]  time = 0.004, size = 31, normalized size = 15.5 \[{1\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34714, size = 19, normalized size = 9.5 \[ \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.225712, size = 24, normalized size = 12. \[ -\log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sqrt(x + 1)*sqrt(x - 1) - x)

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Sympy [A]  time = 3.81922, size = 41, normalized size = 20.5 \[ \begin{cases} 2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 2 i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*acosh(sqrt(2)*sqrt(x + 1)/2), Abs(x + 1)/2 > 1), (-2*I*asin(sqrt(2)
*sqrt(x + 1)/2), True))

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GIAC/XCAS [A]  time = 0.247296, size = 23, normalized size = 11.5 \[ -2 \,{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*ln(abs(-sqrt(x + 1) + sqrt(x - 1)))