\(\int (1-\text {csch}^2(x))^{3/2} \, dx\) [19]

Optimal result

Integrand size = 12, antiderivative size = 46 \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=2 \arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4213, 427, 537, 222, 385, 212} \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=2 \arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)} \]

[In]

Int[(1 - Csch[x]^2)^(3/2),x]

[Out]

2*ArcSin[Coth[x]/Sqrt[2]] + ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]] + (Coth[x]*Sqrt[2 - Coth[x]^2])/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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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 {\left (2-x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right ) \\ & = \frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-6+4 x^2}{\left (1-x^2\right ) \sqrt {2-x^2}} \, dx,x,\coth (x)\right ) \\ & = \frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)}+2 \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 ) \\ & = 2 \arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \\ & = 2 \arcsin \left (\frac {\coth (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.00 \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {\left (1-\text {csch}^2(x)\right )^{3/2} \left (\sqrt {-3+\cosh (2 x)} \coth (x) \text {csch}(x)+2 \sqrt {2} \left (2 \arctan \left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right )\right ) \sinh ^3(x)}{(-3+\cosh (2 x))^{3/2}} \]

[In]

Integrate[(1 - Csch[x]^2)^(3/2),x]

[Out]

((1 - Csch[x]^2)^(3/2)*(Sqrt[-3 + Cosh[2*x]]*Coth[x]*Csch[x] + 2*Sqrt[2]*(2*ArcTan[(Sqrt[2]*Cosh[x])/Sqrt[-3 +
Cosh[2*x]]] + Log[Sqrt[2]*Cosh[x] + Sqrt[-3 + Cosh[2*x]]]))*Sinh[x]^3)/(-3 + Cosh[2*x])^(3/2)

Maple [F]

\[\int \left (1-\operatorname {csch}\left (x \right )^{2}\right )^{\frac {3}{2}}d x\]

[In]

int((1-csch(x)^2)^(3/2),x)

[Out]

int((1-csch(x)^2)^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (39) = 78\).

Time = 0.26 (sec) , antiderivative size = 528, normalized size of antiderivative = 11.48 \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((1-csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*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)) - 8
*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - c
osh(x))*sinh(x) + 1)*arctan(-1/2*cosh(x)^2 - cosh(x)*sinh(x) - 1/2*sinh(x)^2 + 1/2*sqrt(2)*sqrt((cosh(x)^2 + s
inh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/2) - 2*cosh(x)^2 - (cosh(x)^4 + 4*cosh(x)*sinh(
x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*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) + (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1
)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*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) + 4*(cosh(x)^3
- cosh(x))*sinh(x) + 1)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(
x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

Sympy [F]

\[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=\int \left (1 - \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((1-csch(x)**2)**(3/2),x)

[Out]

Integral((1 - csch(x)**2)**(3/2), x)

Maxima [F]

\[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=\int { {\left (-\operatorname {csch}\left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((1-csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((-csch(x)^2 + 1)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (39) = 78\).

Time = 0.32 (sec) , antiderivative size = 253, normalized size of antiderivative = 5.50 \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=-4 \, \arctan \left (\frac {1}{2} \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - \frac {1}{2} \, e^{\left (2 \, x\right )} + \frac {1}{2}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {1}{2} \, \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {16 \, {\left ({\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, {\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 5\right )}^{2}} \]

[In]

integrate((1-csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

-4*arctan(1/2*sqrt(e^(4*x) - 6*e^(2*x) + 1) - 1/2*e^(2*x) + 1/2)*sgn(e^(2*x) - 1) - 1/2*log(-sqrt(e^(4*x) - 6*
e^(2*x) + 1) + e^(2*x) + 1)*sgn(e^(2*x) - 1) - 1/2*log(abs(sqrt(e^(4*x) - 6*e^(2*x) + 1) - e^(2*x) + 3))*sgn(e
^(2*x) - 1) + 1/2*log(abs(sqrt(e^(4*x) - 6*e^(2*x) + 1) - e^(2*x) + 1))*sgn(e^(2*x) - 1) + 16*((sqrt(e^(4*x) -
6*e^(2*x) + 1) - e^(2*x))^2*sgn(e^(2*x) - 1) + 2*(sqrt(e^(4*x) - 6*e^(2*x) + 1) - e^(2*x))*sgn(e^(2*x) - 1) +
sgn(e^(2*x) - 1))/((sqrt(e^(4*x) - 6*e^(2*x) + 1) - e^(2*x))^2 + 2*sqrt(e^(4*x) - 6*e^(2*x) + 1) - 2*e^(2*x)
+ 5)^2

Mupad [F(-1)]

Timed out. \[ \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx=\int {\left (1-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{3/2} \,d x \]

[In]

int((1 - 1/sinh(x)^2)^(3/2),x)

[Out]

int((1 - 1/sinh(x)^2)^(3/2), x)