\(\int \frac {1}{(a \text {csch}^3(x))^{5/2}} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=-\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}} \]

[Out]

-26/77*coth(x)/a^2/(a*csch(x)^3)^(1/2)+78/385*cosh(x)*sinh(x)/a^2/(a*csch(x)^3)^(1/2)-26/165*cosh(x)*sinh(x)^3
/a^2/(a*csch(x)^3)^(1/2)+2/15*cosh(x)*sinh(x)^5/a^2/(a*csch(x)^3)^(1/2)+26/77*I*csch(x)^2*(sin(1/4*Pi+1/2*I*x)
^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2))*(I*sinh(x))^(1/2)/a^2/(a*csch(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2720} \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=-\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \sqrt {i \sinh (x)} \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )}{77 a^2 \sqrt {a \text {csch}^3(x)}} \]

[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 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))^{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) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},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] (verified)

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \text {csch}^3(x)} \sinh (x) \left (24960 i \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right ) \sqrt {i \sinh (x)}-19122 \sinh (2 x)+4406 \sinh (4 x)-826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]

[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]

\[\int \frac {1}{\left (a \operatorname {csch}\left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.32 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/147840*(49920*sqrt(2)*(cosh(x)^8 + 8*cosh(x)^7*sinh(x) + 28*cosh(x)^6*sinh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 7
0*cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8)*sqr
t(a)*weierstrassPInverse(4, 0, cosh(x) + sinh(x)) + sqrt(2)*(77*cosh(x)^16 + 1232*cosh(x)*sinh(x)^15 + 77*sinh
(x)^16 + 14*(660*cosh(x)^2 - 59)*sinh(x)^14 - 826*cosh(x)^14 + 196*(220*cosh(x)^3 - 59*cosh(x))*sinh(x)^13 + 2
*(70070*cosh(x)^4 - 37583*cosh(x)^2 + 2203)*sinh(x)^12 + 4406*cosh(x)^12 + 8*(42042*cosh(x)^5 - 37583*cosh(x)^
3 + 6609*cosh(x))*sinh(x)^11 + 2*(308308*cosh(x)^6 - 413413*cosh(x)^4 + 145398*cosh(x)^2 - 9561)*sinh(x)^10 -
19122*cosh(x)^10 + 4*(220220*cosh(x)^7 - 413413*cosh(x)^5 + 242330*cosh(x)^3 - 47805*cosh(x))*sinh(x)^9 + 6*(1
65165*cosh(x)^8 - 413413*cosh(x)^6 + 363495*cosh(x)^4 - 143415*cosh(x)^2)*sinh(x)^8 + 16*(55055*cosh(x)^9 - 17
7177*cosh(x)^7 + 218097*cosh(x)^5 - 143415*cosh(x)^3)*sinh(x)^7 + 2*(308308*cosh(x)^10 - 1240239*cosh(x)^8 + 2
035572*cosh(x)^6 - 2007810*cosh(x)^4 + 9561)*sinh(x)^6 + 19122*cosh(x)^6 + 4*(84084*cosh(x)^11 - 413413*cosh(x
)^9 + 872388*cosh(x)^7 - 1204686*cosh(x)^5 + 28683*cosh(x))*sinh(x)^5 + 2*(70070*cosh(x)^12 - 413413*cosh(x)^1
0 + 1090485*cosh(x)^8 - 2007810*cosh(x)^6 + 143415*cosh(x)^2 - 2203)*sinh(x)^4 - 4406*cosh(x)^4 + 8*(5390*cosh
(x)^13 - 37583*cosh(x)^11 + 121165*cosh(x)^9 - 286830*cosh(x)^7 + 47805*cosh(x)^3 - 2203*cosh(x))*sinh(x)^3 +
2*(4620*cosh(x)^14 - 37583*cosh(x)^12 + 145398*cosh(x)^10 - 430245*cosh(x)^8 + 143415*cosh(x)^4 - 13218*cosh(x
)^2 + 413)*sinh(x)^2 + 826*cosh(x)^2 + 4*(308*cosh(x)^15 - 2891*cosh(x)^13 + 13218*cosh(x)^11 - 47805*cosh(x)^
9 + 28683*cosh(x)^5 - 4406*cosh(x)^3 + 413*cosh(x))*sinh(x) - 77)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*
cosh(x)*sinh(x) + sinh(x)^2 - 1)))/(a^3*cosh(x)^8 + 8*a^3*cosh(x)^7*sinh(x) + 28*a^3*cosh(x)^6*sinh(x)^2 + 56*
a^3*cosh(x)^5*sinh(x)^3 + 70*a^3*cosh(x)^4*sinh(x)^4 + 56*a^3*cosh(x)^3*sinh(x)^5 + 28*a^3*cosh(x)^2*sinh(x)^6
+ 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8)

Sympy [F]

\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((a*csch(x)**3)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csch(x)^3)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*csch(x)^3)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^3}\right )}^{5/2}} \,d x \]

[In]

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

[Out]

int(1/(a/sinh(x)^3)^(5/2), x)