\(\int (-1+\text {csch}^2(x))^{3/2} \, dx\) [22]

Optimal result

Integrand size = 10, antiderivative size = 47 \[ \int \left (-1+\text {csch}^2(x)\right )^{3/2} \, dx=\arctan \left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )+2 \text {arctanh}\left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {-2+\coth ^2(x)} \]

[Out]

arctan(coth(x)/(-2+coth(x)^2)^(1/2))+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 = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4213, 427, 537, 223, 212, 385, 209} \[ \int \left (-1+\text {csch}^2(x)\right )^{3/2} \, dx=\arctan \left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )+2 \text {arctanh}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )-\frac {1}{2} \coth (x) \sqrt {\coth ^2(x)-2} \]

[In]

Int[(-1 + Csch[x]^2)^(3/2),x]

[Out]

ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]] + 2*ArcTanh[Coth[x]/Sqrt[-2 + Coth[x]^2]] - (Coth[x]*Sqrt[-2 + Coth[x]^2]
)/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 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 ) \\ & = -\frac {1}{2} \coth (x) \sqrt {-2+\coth ^2(x)}+2 \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 )+2 \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 = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.91 \[ \int \left (-1+\text {csch}^2(x)\right )^{3/2} \, dx=\frac {\left (-1+\text {csch}^2(x)\right )^{3/2} \left (\sqrt {-3+\cosh (2 x)} \coth (x) \text {csch}(x)+2 \sqrt {2} \left (2 \arctan \left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right )\right ) \sinh ^3(x)}{(-3+\cosh (2 x))^{3/2}} \]

[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. 666 vs. \(2 (39) = 78\).

Time = 0.26 (sec) , antiderivative size = 666, normalized size of antiderivative = 14.17 \[ \int \left (-1+\text {csch}^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(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 - 1)*sinh(x)^2
- 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*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)*si
nh(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)) +
(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 - co
sh(x))*sinh(x) + 1)*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)) - 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)^
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)) + 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)^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)))/(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]

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

[In]

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

[Out]

Integral((csch(x)**2 - 1)**(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. 412 vs. \(2 (39) = 78\).

Time = 0.39 (sec) , antiderivative size = 412, normalized size of antiderivative = 8.77 \[ \int \left (-1+\text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \, \arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - \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 ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\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 ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\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 ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {2 \, {\left (\sqrt {2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {5 \, \sqrt {2} {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{2}} + \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{3} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{3}} + \frac {5 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{e^{\left (2 \, x\right )} - 3}\right )}}{{\left (\frac {2 \, \sqrt {2} {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3} + \frac {{\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{2}}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{2}} + 1\right )}^{2}} \]

[In]

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

[Out]

-1/2*arcsin(1/4*sqrt(2)*(e^(2*x) - 3))*sgn(-e^(2*x) + 1) - 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) + 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) - 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) + 2*(sqrt(2)*sgn(-e^(2*x) + 1) + 5*sqrt(2)*(2*sqrt(2) - sqrt(-e^(4*x)
+ 6*e^(2*x) - 1))^2*sgn(-e^(2*x) + 1)/(e^(2*x) - 3)^2 + 3*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))^3*sgn(
-e^(2*x) + 1)/(e^(2*x) - 3)^3 + 5*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))*sgn(-e^(2*x) + 1)/(e^(2*x) - 3)
)/(2*sqrt(2)*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3) + (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*
x) - 1))^2/(e^(2*x) - 3)^2 + 1)^2

Mupad [F(-1)]

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

[In]

int((1/sinh(x)^2 - 1)^(3/2),x)

[Out]

int((1/sinh(x)^2 - 1)^(3/2), x)