\(\int \sqrt {a \text {csch}^3(x)} \, dx\) [38]

Optimal result

Integrand size = 10, antiderivative size = 56 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 i \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) (i \sinh (x))^{3/2}-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x) \]

[Out]

-2*I*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticE(cos(1/4*Pi+1/2*I*x),2^(1/2))*(I*sinh(x))^(3/2
)*(a*csch(x)^3)^(1/2)-2*cosh(x)*sinh(x)*(a*csch(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)}}{\sqrt {i \sinh (x)}} \]

[In]

Int[Sqrt[a*Csch[x]^3],x]

[Out]

-2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x] + ((2*I)*Sqrt[a*Csch[x]^3]*EllipticE[Pi/4 - (I/2)*x, 2]*Sinh[x]^2)/Sqrt[I
*Sinh[x]]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \text {csch}^3(x)} \int (i \text {csch}(x))^{3/2} \, dx}{(i \text {csch}(x))^{3/2}} \\ & = -2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {\sqrt {a \text {csch}^3(x)} \int \frac {1}{\sqrt {i \text {csch}(x)}} \, dx}{(i \text {csch}(x))^{3/2}} \\ & = -2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)+\frac {\left (\sqrt {a \text {csch}^3(x)} \sinh ^2(x)\right ) \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)}} \\ & = -2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)+\frac {2 i \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{\sqrt {i \sinh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \sqrt {a \text {csch}^3(x)} \left (\cosh (x)-E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}\right ) \sinh (x) \]

[In]

Integrate[Sqrt[a*Csch[x]^3],x]

[Out]

-2*Sqrt[a*Csch[x]^3]*(Cosh[x] - EllipticE[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*Sinh[x]

Maple [F]

\[\int \sqrt {a \operatorname {csch}\left (x \right )^{3}}d x\]

[In]

int((a*csch(x)^3)^(1/2),x)

[Out]

int((a*csch(x)^3)^(1/2),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \, \sqrt {2} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \]

[In]

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

[Out]

-2*sqrt(2)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1))*(cosh(x) + sinh(x)) -
2*sqrt(2)*sqrt(a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x)))

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a*csch(x)**3), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(a*csch(x)^3), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(a*csch(x)^3), x)

Mupad [F(-1)]

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

[In]

int((a/sinh(x)^3)^(1/2),x)

[Out]

int((a/sinh(x)^3)^(1/2), x)