\(\int \sqrt {1-\text {csch}^2(x)} \, dx\) [20]

Optimal result

Integrand size = 12, antiderivative size = 26 \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4213, 399, 222, 385, 212} \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \]

[In]

Int[Sqrt[1 - Csch[x]^2],x]

[Out]

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

Rule 212

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

Rule 222

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 385

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 399

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

Rule 4213

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).

Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\frac {\sqrt {2-2 \text {csch}^2(x)} \left (\arctan \left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right ) \sinh (x)}{\sqrt {-3+\cosh (2 x)}} \]

[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]]

Maple [F]

\[\int \sqrt {1-\operatorname {csch}\left (x \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (23) = 46\).

Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 8.50 \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=-2 \, \arctan \left (-\frac {1}{2} \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right ) - \frac {1}{2} \, \sinh \left (x\right )^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right ) + \frac {1}{2} \, \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt {2} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1\right ) \]

[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)

Sympy [F]

\[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\int \sqrt {1 - \operatorname {csch}^{2}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(1 - csch(x)**2), x)

Maxima [F]

\[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\int { \sqrt {-\operatorname {csch}\left (x\right )^{2} + 1} \,d x } \]

[In]

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

[Out]

integrate(sqrt(-csch(x)^2 + 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (23) = 46\).

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.23 \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=-\frac {1}{2} \, {\left (4 \, \arctan \left (\frac {1}{2} \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - \frac {1}{2} \, e^{\left (2 \, x\right )} + \frac {1}{2}\right ) + \log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) + \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) - \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1-\text {csch}^2(x)} \, dx=\int \sqrt {1-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}} \,d x \]

[In]

int((1 - 1/sinh(x)^2)^(1/2),x)

[Out]

int((1 - 1/sinh(x)^2)^(1/2), x)