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

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

[Out]

ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]]

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Rubi [A]  time = 0.0178882, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4128, 377, 203} \[ \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{\coth ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + Csch[x]^2],x]

[Out]

ArcTan[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 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 203

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

Rubi steps

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

Mathematica [B]  time = 0.0302294, size = 48, normalized size = 3.43 \[ \frac{\sqrt{\cosh (2 x)-3} \text{csch}(x) \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)-3}\right )}{\sqrt{2} \sqrt{\text{csch}^2(x)-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + Csch[x]^2],x]

[Out]

(Sqrt[-3 + Cosh[2*x]]*Csch[x]*Log[Sqrt[2]*Cosh[x] + Sqrt[-3 + Cosh[2*x]]])/(Sqrt[2]*Sqrt[-1 + Csch[x]^2])

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Maple [F]  time = 0.14, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\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/(-1+csch(x)^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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/(-1+csch(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(csch(x)^2 - 1), x)

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Fricas [B]  time = 2.20857, size = 756, normalized size = 54. \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\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}}}}{\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} + 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} \, \arctan \left (\frac{\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}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \,{\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(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))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 2)*sinh(x
)^2 + 4*cosh(x)^2 + 4*(cosh(x)^3 + 2*cosh(x))*sinh(x) - 1)) - 1/2*arctan(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))/(cosh(x)^4 +
4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(
x) + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(csch(x)**2 - 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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/(-1+csch(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(csch(x)^2 - 1), x)