3.20 \(\int \sqrt{1-\text{csch}^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )+\sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right ) \]

[Out]

ArcSin[Coth[x]/Sqrt[2]] + ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]]

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Rubi [A]  time = 0.0241561, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4128, 402, 216, 377, 206} \[ \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )+\sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Csch[x]^2],x]

[Out]

ArcSin[Coth[x]/Sqrt[2]] + ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]]

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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

Mathematica [B]  time = 0.0639665, size = 65, normalized size = 2.5 \[ \frac{\sinh (x) \sqrt{2-2 \text{csch}^2(x)} \left (\log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)-3}\right )+\tan ^{-1}\left (\frac{\sqrt{2} \cosh (x)}{\sqrt{\cosh (2 x)-3}}\right )\right )}{\sqrt{\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Csch[x]^2],x]

[Out]

(Sqrt[2 - 2*Csch[x]^2]*(ArcTan[(Sqrt[2]*Cosh[x])/Sqrt[-3 + Cosh[2*x]]] + Log[Sqrt[2]*Cosh[x] + Sqrt[-3 + Cosh[
2*x]]])*Sinh[x])/Sqrt[-3 + Cosh[2*x]]

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \sqrt{1- \left ({\rm csch} \left (x\right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\operatorname{csch}\left (x\right )^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-csch(x)^2 + 1), x)

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Fricas [B]  time = 2.21966, size = 786, normalized size = 30.23 \begin{align*} -2 \, \arctan \left (-\frac{1}{2} \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right ) - \frac{1}{2} \, \sinh \left (x\right )^{2} + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - \sqrt{2}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right ) + \frac{1}{2} \, \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt{2} \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(-1/2*cosh(x)^2 - cosh(x)*sinh(x) - 1/2*sinh(x)^2 + 1/2*sqrt(2)*sqrt((cosh(x)^2 + sinh(x)^2 - 3)/(cos
h(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/2) - 1/2*log(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*c
osh(x)^2 - 2)*sinh(x)^2 - sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt((cosh(x)^2 + sinh(x)^2
- 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*cosh(x)^2 + 4*(cosh(x)^3 - 2*cosh(x))*sinh(x) - 1) + 1/2
*log(-cosh(x)^2 - 2*cosh(x)*sinh(x) - sinh(x)^2 + sqrt(2)*sqrt((cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh
(x)*sinh(x) + sinh(x)^2)) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - csch(x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\operatorname{csch}\left (x\right )^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-csch(x)^2 + 1), x)