\(\int \sqrt {-1+\text {csch}^2(x)} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 33 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=-\arctan \left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4213, 399, 223, 212, 385, 209} \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=-\arctan \left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )-\text {arctanh}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]

[In]

Int[Sqrt[-1 + Csch[x]^2],x]

[Out]

-ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]] - ArcTanh[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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 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 ) \\ & = -\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-2+\coth ^2(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 )-\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. \(68\) vs. \(2(33)=66\).

Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.06 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\frac {\sqrt {2} \sqrt {-1+\text {csch}^2(x)} \left (\arctan \left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right ) \sinh (x)}{\sqrt {-3+\cosh (2 x)}} \]

[In]

Integrate[Sqrt[-1 + Csch[x]^2],x]

[Out]

(Sqrt[2]*Sqrt[-1 + 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. 358 vs. \(2 (29) = 58\).

Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 10.85 \[ \int \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 ) - \frac {1}{2} \, \log \left (\frac {\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}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {\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}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) \]

[In]

integrate((-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)) - 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)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 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)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(csch(x)**2 - 1), 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. 157 vs. \(2 (29) = 58\).

Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.76 \[ \int \sqrt {-1+\text {csch}^2(x)} \, dx=\frac {1}{2} \, {\left (\arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) + 2 \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right ) - 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} + 1 \right |}\right ) + 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((1/sinh(x)^2 - 1)^(1/2),x)

[Out]

int((1/sinh(x)^2 - 1)^(1/2), x)