\(\int (a \text {csch}^2(x))^{3/2} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \]

[Out]

1/2*a^(3/2)*arctanh(coth(x)*a^(1/2)/(a*csch(x)^2)^(1/2))-1/2*a*coth(x)*(a*csch(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223, 212} \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \]

[In]

Int[(a*Csch[x]^2)^(3/2),x]

[Out]

(a^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]])/2 - (a*Coth[x]*Sqrt[a*Csch[x]^2])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps \begin{align*} \text {integral}& = -\left (a \text {Subst}\left (\int \sqrt {-a+a x^2} \, dx,x,\coth (x)\right )\right ) \\ & = -\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)}+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {-a+a x^2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)}+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a \text {csch}^2(x)}}\right ) \\ & = \frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} a \sqrt {a \text {csch}^2(x)} \left (\coth (x) \text {csch}(x)-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x) \]

[In]

Integrate[(a*Csch[x]^2)^(3/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs. \(2(34)=68\).

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.24

method	result	size
risch	\(-\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}+\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right )}{2}-\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{2}\)	\(103\)

[In]

int((a*csch(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-a/(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)*(1+exp(2*x))+1/2*a*exp(-x)*(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x
)-1)^2)^(1/2)*ln(exp(x)+1)-1/2*a*exp(-x)*(exp(2*x)-1)*(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)*ln(exp(x)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (34) = 68\).

Time = 0.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 7.39 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=\frac {{\left (2 \, a \cosh \left (x\right )^{3} - 2 \, {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{3} - 6 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) - 2 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - {\left (a \cosh \left (x\right )^{4} - {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{4} - 4 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} - {\left (3 \, a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{2} - {\left (a \cosh \left (x\right )^{4} - 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right ) - {\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} - {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )}} \]

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(2*a*cosh(x)^3 - 2*(a*e^(2*x) - a)*sinh(x)^3 - 6*(a*cosh(x)*e^(2*x) - a*cosh(x))*sinh(x)^2 + 2*a*cosh(x) -
2*(a*cosh(x)^3 + a*cosh(x))*e^(2*x) - (a*cosh(x)^4 - (a*e^(2*x) - a)*sinh(x)^4 - 4*(a*cosh(x)*e^(2*x) - a*cos
h(x))*sinh(x)^3 - 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 - (3*a*cosh(x)^2 - a)*e^(2*x) - a)*sinh(x)^2 - (a*cosh(x)^4
- 2*a*cosh(x)^2 + a)*e^(2*x) + 4*(a*cosh(x)^3 - a*cosh(x) - (a*cosh(x)^3 - a*cosh(x))*e^(2*x))*sinh(x) + a)*l
og((cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x) - 1)) + 2*(3*a*cosh(x)^2 - (3*a*cosh(x)^2 + a)*e^(2*x) + a)*sinh
(x))*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*e^x/(4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*e^x*
sinh(x)^2 + 4*(cosh(x)^3 - cosh(x))*e^x*sinh(x) + (cosh(x)^4 - 2*cosh(x)^2 + 1)*e^x)

Sympy [F]

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

[In]

integrate((a*csch(x)**2)**(3/2),x)

[Out]

Integral((a*csch(x)**2)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} - 1\right ) - \frac {a^{\frac {3}{2}} e^{\left (-x\right )} + a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*a^(3/2)*log(e^(-x) + 1) + 1/2*a^(3/2)*log(e^(-x) - 1) - (a^(3/2)*e^(-x) + a^(3/2)*e^(-3*x))/(2*e^(-2*x) -
e^(-4*x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, a^{\frac {3}{2}} {\left (\frac {4 \, {\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/4*a^(3/2)*(4*(e^(-x) + e^x)/((e^(-x) + e^x)^2 - 4) - log(e^(-x) + e^x + 2) + log(e^(-x) + e^x - 2))*sgn(e^(
3*x) - e^x)

Mupad [F(-1)]

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

[In]

int((a/sinh(x)^2)^(3/2),x)

[Out]

int((a/sinh(x)^2)^(3/2), x)