3.11 \(\int \sqrt{-1+\coth ^2(x)} \, dx\)

Optimal. Leaf size=14 \[ -\tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}\right ) \]

[Out]

-ArcTanh[Coth[x]/Sqrt[Csch[x]^2]]

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Rubi [A]  time = 0.0214563, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3657, 4122, 217, 206} \[ -\tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{\text{csch}^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Coth[x]^2],x]

[Out]

-ArcTanh[Coth[x]/Sqrt[Csch[x]^2]]

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

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

Mathematica [A]  time = 0.0065335, size = 18, normalized size = 1.29 \[ \sinh (x) \sqrt{\text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Coth[x]^2],x]

[Out]

Sqrt[Csch[x]^2]*Log[Tanh[x/2]]*Sinh[x]

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Maple [A]  time = 0.045, size = 15, normalized size = 1.1 \begin{align*} -\ln \left ({\rm coth} \left (x\right )+\sqrt{-1+ \left ({\rm coth} \left (x\right ) \right ) ^{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.73775, size = 23, normalized size = 1.64 \begin{align*} \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e^(-x) + 1) - log(e^(-x) - 1)

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Fricas [A]  time = 2.50292, size = 78, normalized size = 5.57 \begin{align*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(cosh(x) + sinh(x) + 1) + log(cosh(x) + sinh(x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\coth ^{2}{\left (x \right )} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(coth(x)**2 - 1), x)

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Giac [A]  time = 1.12787, size = 31, normalized size = 2.21 \begin{align*} -{\left (\log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(e^x + 1) - log(abs(e^x - 1)))*sgn(e^(2*x) - 1)