\(\int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 14 \[ \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx=\arctan \left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right ) \]

[Out]

arctan(coth(x)/(-2+coth(x)^2)^(1/2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4213, 385, 209} \[ \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx=\arctan \left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]

[In]

Int[1/Sqrt[-1 + Csch[x]^2],x]

[Out]

ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 ) \\ & = \arctan \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. \(48\) vs. \(2(14)=28\).

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.43 \[ \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx=\frac {\sqrt {-3+\cosh (2 x)} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )}{\sqrt {2} \sqrt {-1+\text {csch}^2(x)}} \]

[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]*Sqrt[-1 + 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. 213 vs. \(2 (12) = 24\).

Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 15.21 \[ \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) \]

[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))

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(csch(x)**2 - 1), 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. 69 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.93 \[ \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx=-\frac {\arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) + 2 \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right )}{2 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(1/sinh(x)^2 - 1)^(1/2),x)

[Out]

int(1/(1/sinh(x)^2 - 1)^(1/2), x)