\(\int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx\) [21]

Optimal result

Integrand size = 12, antiderivative size = 16 \[ \int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx=\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4213, 385, 212} \[ \int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx=\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \]

[In]

Int[1/Sqrt[1 - Csch[x]^2],x]

[Out]

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 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 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 {1}{\left (1-x^2\right ) \sqrt {2-x^2}} \, dx,x,\coth (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\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. \(45\) vs. \(2(16)=32\).

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.81 \[ \int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx=\frac {\sqrt {-3+\cosh (2 x)} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )}{\sqrt {2-2 \text {csch}^2(x)}} \]

[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 - 2*Csch[x]^2]

Maple [F]

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

[In]

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

[Out]

int(1/(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. 163 vs. \(2 (14) = 28\).

Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 10.19 \[ \int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx=-\frac {1}{2} \, \log \left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right ) + \frac {1}{2} \, \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt {2} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1\right ) \]

[In]

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

[Out]

-1/2*log(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(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]

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

[In]

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

[Out]

Integral(1/sqrt(1 - csch(x)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/sqrt(-csch(x)^2 + 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 5.31 \[ \int \frac {1}{\sqrt {1-\text {csch}^2(x)}} \, dx=-\frac {\log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) + \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) - \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right )}{2 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]

[In]

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

[Out]

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

[In]

int(1/(1 - 1/sinh(x)^2)^(1/2),x)

[Out]

int(1/(1 - 1/sinh(x)^2)^(1/2), x)