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

Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]

[Out]

ArcSin[Coth[x]]

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Rubi [A]  time = 0.0183658, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 216} \[ \sin ^{-1}(\coth (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Coth[x]^2],x]

[Out]

ArcSin[Coth[x]]

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 216

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

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 )\\ &=\sin ^{-1}(\coth (x))\\ \end{align*}

Mathematica [B]  time = 0.0064008, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt{-\text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

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 = 0.037, size = 4, normalized size = 1.3 \begin{align*} \arcsin \left ({\rm coth} \left (x\right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(coth(x))

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Maxima [C]  time = 1.72328, size = 26, normalized size = 8.67 \begin{align*} i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \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]

I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

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Fricas [A]  time = 2.61366, size = 4, normalized size = 1.33 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

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Giac [C]  time = 1.16479, size = 35, normalized size = 11.67 \begin{align*}{\left (i \, \log \left (e^{x} + 1\right ) - i \, \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]

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