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

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

[Out]

ArcCosh[x]

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Rubi [A]  time = 0.0011083, 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, Rules used = {52} \[ \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*Sqrt[1 + x]),x]

[Out]

ArcCosh[x]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx &=\cosh ^{-1}(x)\\ \end{align*}

Mathematica [B]  time = 0.0022674, size = 18, normalized size = 9. \[ 2 \tanh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTanh[Sqrt[-1 + x]/Sqrt[1 + x]]

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Maple [B]  time = 0.002, size = 31, normalized size = 15.5 \begin{align*}{\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}}}} \end{align*}

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 [B]  time = 1.06832, size = 19, normalized size = 9.5 \begin{align*} \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

log(2*x + 2*sqrt(x^2 - 1))

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Fricas [B]  time = 1.52784, size = 47, normalized size = 23.5 \begin{align*} -\log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

-log(sqrt(x + 1)*sqrt(x - 1) - x)

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Sympy [B]  time = 1.33179, size = 41, normalized size = 20.5 \begin{align*} \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} \end{align*}

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 [B]  time = 1.29858, size = 23, normalized size = 11.5 \begin{align*} -2 \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

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