3.41 \(\int \frac{1}{(a \text{csch}^3(x))^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{26 i \sqrt{i \sinh (x)} \text{csch}^2(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right )}{77 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \coth (x)}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{78 \sinh (x) \cosh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}} \]

[Out]

(-26*Coth[x])/(77*a^2*Sqrt[a*Csch[x]^3]) + (((26*I)/77)*Csch[x]^2*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]
)/(a^2*Sqrt[a*Csch[x]^3]) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqrt[a*Csch[x]^3]) - (26*Cosh[x]*Sinh[x]^3)/(165*a^2
*Sqrt[a*Csch[x]^3]) + (2*Cosh[x]*Sinh[x]^5)/(15*a^2*Sqrt[a*Csch[x]^3])

________________________________________________________________________________________

Rubi [A]  time = 0.0707349, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ -\frac{26 \coth (x)}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{78 \sinh (x) \cosh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{26 i \sqrt{i \sinh (x)} \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{77 a^2 \sqrt{a \text{csch}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^3)^(-5/2),x]

[Out]

(-26*Coth[x])/(77*a^2*Sqrt[a*Csch[x]^3]) + (((26*I)/77)*Csch[x]^2*EllipticF[Pi/4 - (I/2)*x, 2]*Sqrt[I*Sinh[x]]
)/(a^2*Sqrt[a*Csch[x]^3]) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqrt[a*Csch[x]^3]) - (26*Cosh[x]*Sinh[x]^3)/(165*a^2
*Sqrt[a*Csch[x]^3]) + (2*Cosh[x]*Sinh[x]^5)/(15*a^2*Sqrt[a*Csch[x]^3])

Rule 4123

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]

Rule 3769

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 3771

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 2641

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a \text{csch}^3(x)\right )^{5/2}} \, dx &=-\frac{(i \text{csch}(x))^{3/2} \int \frac{1}{(i \text{csch}(x))^{15/2}} \, dx}{a^2 \sqrt{a \text{csch}^3(x)}}\\ &=\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{\left (13 (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{(i \text{csch}(x))^{11/2}} \, dx}{15 a^2 \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{\left (39 (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{(i \text{csch}(x))^{7/2}} \, dx}{55 a^2 \sqrt{a \text{csch}^3(x)}}\\ &=\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{\left (39 (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{(i \text{csch}(x))^{3/2}} \, dx}{77 a^2 \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{26 \coth (x)}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{\left (13 (i \text{csch}(x))^{3/2}\right ) \int \sqrt{i \text{csch}(x)} \, dx}{77 a^2 \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{26 \coth (x)}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{\left (13 \text{csch}^2(x) \sqrt{i \sinh (x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx}{77 a^2 \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{26 \coth (x)}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{26 i \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)}}{77 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{csch}^3(x)}}-\frac{26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt{a \text{csch}^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.120328, size = 71, normalized size = 0.53 \[ \frac{\sinh (x) \sqrt{a \text{csch}^3(x)} \left (24960 i \sqrt{i \sinh (x)} \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )-19122 \sinh (2 x)+4406 \sinh (4 x)-826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^3)^(-5/2),x]

[Out]

(Sqrt[a*Csch[x]^3]*Sinh[x]*((24960*I)*EllipticF[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]] - 19122*Sinh[2*x] + 4406*
Sinh[4*x] - 826*Sinh[6*x] + 77*Sinh[8*x]))/(73920*a^3)

________________________________________________________________________________________

Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm csch} \left (x\right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csch(x)^3)^(5/2),x)

[Out]

int(1/(a*csch(x)^3)^(5/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*csch(x)^3)^(-5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \operatorname{csch}\left (x\right )^{3}}}{a^{3} \operatorname{csch}\left (x\right )^{9}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*csch(x)^3)/(a^3*csch(x)^9), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}^{3}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)**3)**(5/2),x)

[Out]

Integral((a*csch(x)**3)**(-5/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csch(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*csch(x)^3)^(-5/2), x)