\(\int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx\) [39]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{3 \sqrt {a \text {csch}^3(x)}} \]

[Out]

2/3*coth(x)/(a*csch(x)^3)^(1/2)-2/3*I*csch(x)^2*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(co
s(1/4*Pi+1/2*I*x),2^(1/2))*(I*sinh(x))^(1/2)/(a*csch(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, 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 = {4208, 3854, 3856, 2720} \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \sqrt {i \sinh (x)} \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )}{3 \sqrt {a \text {csch}^3(x)}} \]

[In]

Int[1/Sqrt[a*Csch[x]^3],x]

[Out]

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

Rule 2720

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

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 {(i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{3/2}} \, dx}{\sqrt {a \text {csch}^3(x)}} \\ & = \frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}+\frac {(i \text {csch}(x))^{3/2} \int \sqrt {i \text {csch}(x)} \, dx}{3 \sqrt {a \text {csch}^3(x)}} \\ & = \frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {\left (\text {csch}^2(x) \sqrt {i \sinh (x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{3 \sqrt {a \text {csch}^3(x)}} \\ & = \frac {2 \coth (x)}{3 \sqrt {a \text {csch}^3(x)}}-\frac {2 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{3 \sqrt {a \text {csch}^3(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\frac {2 \left (\coth (x)+\frac {\text {csch}(x) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right )}{3 \sqrt {a \text {csch}^3(x)}} \]

[In]

Integrate[1/Sqrt[a*Csch[x]^3],x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

int(1/(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 = 127, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=-\frac {4 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - \sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{6 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2}\right )}} \]

[In]

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

[Out]

-1/6*(4*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(a)*weierstrassPInverse(4, 0, cosh(x) + sinh(x
)) - sqrt(2)*(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)*s
qrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(
x) + a*sinh(x)^2)

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a*csch(x)**3), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {csch}\left (x\right )^{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(a*csch(x)^3), x)

Giac [F]

\[ \int \frac {1}{\sqrt {a \text {csch}^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \operatorname {csch}\left (x\right )^{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/sqrt(a*csch(x)^3), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a/sinh(x)^3)^(1/2), x)