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(1/2*coth(x)*2^(1/2))+arctanh(coth(x)/(2-coth(x)^2)^(1/2))

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 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])

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 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 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]

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*}

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Mathematica [B]  time = 0.07, size = 65, normalized size = 2.50 \[ \frac {\sinh (x) \sqrt {2-2 \text {csch}^2(x)} \left (\log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)-3}\right )+\tan ^{-1}\left (\frac {\sqrt {2} \cosh (x)}{\sqrt {\cosh (2 x)-3}}\right )\right )}{\sqrt {\cosh (2 x)-3}} \]

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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fricas [B]  time = 0.44, size = 221, normalized size = 8.50 \[ -2 \, \arctan \left (-\frac {1}{2} \, \cosh \relax (x)^{2} - \cosh \relax (x) \sinh \relax (x) - \frac {1}{2} \, \sinh \relax (x)^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 2\right )} \sinh \relax (x)^{2} - \sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 2 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1\right ) + \frac {1}{2} \, \log \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + \sqrt {2} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1\right ) \]

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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giac [B]  time = 0.15, size = 110, normalized size = 4.23 \[ -\frac {1}{2} \, {\left (4 \, \arctan \left (\frac {1}{2} \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - \frac {1}{2} \, e^{\left (2 \, x\right )} + \frac {1}{2}\right ) + \log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) + \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) - \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*arctan(1/2*sqrt(e^(4*x) - 6*e^(2*x) + 1) - 1/2*e^(2*x) + 1/2) + log(-sqrt(e^(4*x) - 6*e^(2*x) + 1) + e
^(2*x) + 1) + log(abs(sqrt(e^(4*x) - 6*e^(2*x) + 1) - e^(2*x) + 3)) - log(abs(sqrt(e^(4*x) - 6*e^(2*x) + 1) -
e^(2*x) + 1)))*sgn(e^(2*x) - 1)

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maple [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \sqrt {1-\mathrm {csch}\relax (x )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\operatorname {csch}\relax (x)^{2} + 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {1-\frac {1}{{\mathrm {sinh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 1/sinh(x)^2)^(1/2),x)

[Out]

int((1 - 1/sinh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {1 - \operatorname {csch}^{2}{\relax (x )}}\, dx \]

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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