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

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

[Out]

arccosh(x)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, 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 = 0.00, size = 18, normalized size = 9.00 \[ 2 \tanh ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTanh[Sqrt[-1 + x]/Sqrt[1 + x]]

________________________________________________________________________________________

fricas [B]  time = 1.07, size = 18, normalized size = 9.00 \[ -\log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]

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)

________________________________________________________________________________________

giac [B]  time = 1.18, size = 16, normalized size = 8.00 \[ -2 \, \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]

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(sqrt(x + 1) - sqrt(x - 1))

________________________________________________________________________________________

maple [B]  time = 0.00, size = 31, normalized size = 15.50 \[ \frac {\sqrt {\left (x +1\right ) \left (x -1\right )}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x -1}\, \sqrt {x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-1)^(1/2)/(x+1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 14, normalized size = 7.00 \[ \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]

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))

________________________________________________________________________________________

mupad [B]  time = 1.00, size = 2, normalized size = 1.00 \[ \mathrm {acosh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

acosh(x)

________________________________________________________________________________________

sympy [B]  time = 1.05, size = 41, normalized size = 20.50 \[ \begin {cases} 2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________