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

Optimal. Leaf size=12 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

ArcTanh[x/Sqrt[-1 + x^2]]

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Rubi [A]  time = 0.0023553, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-1 + x^2]]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-1+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{-1+x^2}}\right )\\ \end{align*}

Mathematica [B]  time = 0.00269, size = 38, normalized size = 3.17 \[ \frac{1}{2} \log \left (\frac{x}{\sqrt{x^2-1}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + x^2],x]

[Out]

-Log[1 - x/Sqrt[-1 + x^2]]/2 + Log[1 + x/Sqrt[-1 + x^2]]/2

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} \ln \left ( x+\sqrt{{x}^{2}-1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.954707, size = 19, normalized size = 1.58 \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/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 1.60869, size = 35, normalized size = 2.92 \begin{align*} -\log \left (-x + \sqrt{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 - 1))

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Sympy [A]  time = 0.124899, size = 2, normalized size = 0.17 \begin{align*} \operatorname{acosh}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2-1)**(1/2),x)

[Out]

acosh(x)

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Giac [A]  time = 1.09333, size = 20, normalized size = 1.67 \begin{align*} -\log \left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-x + sqrt(x^2 - 1)))