3.1.19 \(\int \frac {1}{\sqrt {-1+x^2}} \, dx\) [19]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {223, 212} \begin {gather*} \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + x^2],x]

[Out]

ArcTanh[x/Sqrt[-1 + x^2]]

Rule 212

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

Rule 223

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
time = 0.00, size = 38, normalized size = 3.17 \begin {gather*} -\frac {1}{2} \log \left (1-\frac {x}{\sqrt {-1+x^2}}\right )+\frac {1}{2} \log \left (1+\frac {x}{\sqrt {-1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + x^2],x]

[Out]

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

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Maple [A]
time = 0.05, size = 11, normalized size = 0.92

method	result	size
default	\(\ln \left (x +\sqrt {x^{2}-1}\right )\)	\(11\)
trager	\(\ln \left (x +\sqrt {x^{2}-1}\right )\)	\(11\)
pseudoelliptic	\(\arctanh \left (\frac {\sqrt {x^{2}-1}}{x}\right )\)	\(13\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \arcsin \left (x \right )}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 14, normalized size = 1.17 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \end {gather*}

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 = 0.58, size = 14, normalized size = 1.17 \begin {gather*} -\log \left (-x + \sqrt {x^{2} - 1}\right ) \end {gather*}

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 = 0.05, size = 10, normalized size = 0.83 \begin {gather*} \log {\left (x + \sqrt {x^{2} - 1} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + sqrt(x**2 - 1))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
time = 0.45, size = 26, normalized size = 2.17 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} x + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.12, size = 10, normalized size = 0.83 \begin {gather*} \ln \left (x+\sqrt {x^2-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 - 1)^(1/2),x)

[Out]

log(x + (x^2 - 1)^(1/2))

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