\(\int (a \text {csch}^3(x))^{5/2} \, dx\) [36]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {154 i a^2 \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 \sqrt {i \sinh (x)}} \]

[Out]

-154/585*a^2*coth(x)*(a*csch(x)^3)^(1/2)+22/117*a^2*coth(x)*csch(x)^2*(a*csch(x)^3)^(1/2)-2/13*a^2*coth(x)*csc
h(x)^4*(a*csch(x)^3)^(1/2)+154/195*a^2*cosh(x)*sinh(x)*(a*csch(x)^3)^(1/2)-154/195*I*a^2*(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))*sinh(x)^2*(a*csch(x)^3)^(1/2)/(I*sinh(x))^
(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, 3853, 3856, 2719} \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {154 i a^2 \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)}}{195 \sqrt {i \sinh (x)}} \]

[In]

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

[Out]

(-154*a^2*Coth[x]*Sqrt[a*Csch[x]^3])/585 + (22*a^2*Coth[x]*Csch[x]^2*Sqrt[a*Csch[x]^3])/117 - (2*a^2*Coth[x]*C
sch[x]^4*Sqrt[a*Csch[x]^3])/13 + (154*a^2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x])/195 - (((154*I)/195)*a^2*Sqrt[a*C
sch[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 {\left (a^2 \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{15/2} \, dx}{(i \text {csch}(x))^{3/2}} \\ & = -\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}-\frac {\left (11 a^2 \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{11/2} \, dx}{13 (i \text {csch}(x))^{3/2}} \\ & = \frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}-\frac {\left (77 a^2 \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{7/2} \, dx}{117 (i \text {csch}(x))^{3/2}} \\ & = -\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}-\frac {\left (77 a^2 \sqrt {a \text {csch}^3(x)}\right ) \int (i \text {csch}(x))^{3/2} \, dx}{195 (i \text {csch}(x))^{3/2}} \\ & = -\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)+\frac {\left (77 a^2 \sqrt {a \text {csch}^3(x)}\right ) \int \frac {1}{\sqrt {i \text {csch}(x)}} \, dx}{195 (i \text {csch}(x))^{3/2}} \\ & = -\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {\left (77 a^2 \sqrt {a \text {csch}^3(x)} \sinh ^2(x)\right ) \int \sqrt {i \sinh (x)} \, dx}{195 \sqrt {i \sinh (x)}} \\ & = -\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {154 i a^2 \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 \sqrt {i \sinh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.50 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=-\frac {2}{585} a^2 \sqrt {a \text {csch}^3(x)} \left (-231 \cosh (x)+\coth (x) \text {csch}(x) \left (77-55 \text {csch}^2(x)+45 \text {csch}^4(x)\right )+231 E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}\right ) \sinh (x) \]

[In]

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

[Out]

(-2*a^2*Sqrt[a*Csch[x]^3]*(-231*Cosh[x] + Coth[x]*Csch[x]*(77 - 55*Csch[x]^2 + 45*Csch[x]^4) + 231*EllipticE[(
Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*Sinh[x])/585

Maple [F]

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

[In]

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

[Out]

int((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 = 1389, normalized size of antiderivative = 10.29 \[ \int \left (a \text {csch}^3(x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/585*(231*sqrt(2)*(a^2*cosh(x)^12 + 12*a^2*cosh(x)*sinh(x)^11 + a^2*sinh(x)^12 - 6*a^2*cosh(x)^10 + 6*(11*a^2
*cosh(x)^2 - a^2)*sinh(x)^10 + 15*a^2*cosh(x)^8 + 20*(11*a^2*cosh(x)^3 - 3*a^2*cosh(x))*sinh(x)^9 + 15*(33*a^2
*cosh(x)^4 - 18*a^2*cosh(x)^2 + a^2)*sinh(x)^8 - 20*a^2*cosh(x)^6 + 24*(33*a^2*cosh(x)^5 - 30*a^2*cosh(x)^3 +
5*a^2*cosh(x))*sinh(x)^7 + 4*(231*a^2*cosh(x)^6 - 315*a^2*cosh(x)^4 + 105*a^2*cosh(x)^2 - 5*a^2)*sinh(x)^6 + 1
5*a^2*cosh(x)^4 + 24*(33*a^2*cosh(x)^7 - 63*a^2*cosh(x)^5 + 35*a^2*cosh(x)^3 - 5*a^2*cosh(x))*sinh(x)^5 + 15*(
33*a^2*cosh(x)^8 - 84*a^2*cosh(x)^6 + 70*a^2*cosh(x)^4 - 20*a^2*cosh(x)^2 + a^2)*sinh(x)^4 - 6*a^2*cosh(x)^2 +
20*(11*a^2*cosh(x)^9 - 36*a^2*cosh(x)^7 + 42*a^2*cosh(x)^5 - 20*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^3 + 6*
(11*a^2*cosh(x)^10 - 45*a^2*cosh(x)^8 + 70*a^2*cosh(x)^6 - 50*a^2*cosh(x)^4 + 15*a^2*cosh(x)^2 - a^2)*sinh(x)^
2 + a^2 + 12*(a^2*cosh(x)^11 - 5*a^2*cosh(x)^9 + 10*a^2*cosh(x)^7 - 10*a^2*cosh(x)^5 + 5*a^2*cosh(x)^3 - a^2*c
osh(x))*sinh(x))*sqrt(a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x))) + sqrt(2)*(231*a^
2*cosh(x)^13 + 3003*a^2*cosh(x)*sinh(x)^12 + 231*a^2*sinh(x)^13 - 1540*a^2*cosh(x)^11 + 154*(117*a^2*cosh(x)^2
- 10*a^2)*sinh(x)^11 + 4367*a^2*cosh(x)^9 + 1694*(39*a^2*cosh(x)^3 - 10*a^2*cosh(x))*sinh(x)^10 + 11*(15015*a
^2*cosh(x)^4 - 7700*a^2*cosh(x)^2 + 397*a^2)*sinh(x)^9 - 6808*a^2*cosh(x)^7 + 33*(9009*a^2*cosh(x)^5 - 7700*a^
2*cosh(x)^3 + 1191*a^2*cosh(x))*sinh(x)^8 + 4*(99099*a^2*cosh(x)^6 - 127050*a^2*cosh(x)^4 + 39303*a^2*cosh(x)^
2 - 1702*a^2)*sinh(x)^7 + 1277*a^2*cosh(x)^5 + 28*(14157*a^2*cosh(x)^7 - 25410*a^2*cosh(x)^5 + 13101*a^2*cosh(
x)^3 - 1702*a^2*cosh(x))*sinh(x)^6 + (297297*a^2*cosh(x)^8 - 711480*a^2*cosh(x)^6 + 550242*a^2*cosh(x)^4 - 142
968*a^2*cosh(x)^2 + 1277*a^2)*sinh(x)^5 - 484*a^2*cosh(x)^3 + (165165*a^2*cosh(x)^9 - 508200*a^2*cosh(x)^7 + 5
50242*a^2*cosh(x)^5 - 238280*a^2*cosh(x)^3 + 6385*a^2*cosh(x))*sinh(x)^4 + 2*(33033*a^2*cosh(x)^10 - 127050*a^
2*cosh(x)^8 + 183414*a^2*cosh(x)^6 - 119140*a^2*cosh(x)^4 + 6385*a^2*cosh(x)^2 - 242*a^2)*sinh(x)^3 + 77*a^2*c
osh(x) + 2*(9009*a^2*cosh(x)^11 - 42350*a^2*cosh(x)^9 + 78606*a^2*cosh(x)^7 - 71484*a^2*cosh(x)^5 + 6385*a^2*c
osh(x)^3 - 726*a^2*cosh(x))*sinh(x)^2 + (3003*a^2*cosh(x)^12 - 16940*a^2*cosh(x)^10 + 39303*a^2*cosh(x)^8 - 47
656*a^2*cosh(x)^6 + 6385*a^2*cosh(x)^4 - 1452*a^2*cosh(x)^2 + 77*a^2)*sinh(x))*sqrt((a*cosh(x) + a*sinh(x))/(c
osh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)))/(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cosh(
x)^2 - 1)*sinh(x)^10 - 6*cosh(x)^10 + 20*(11*cosh(x)^3 - 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 - 18*cosh(x)^
2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 - 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6 -
315*cosh(x)^4 + 105*cosh(x)^2 - 5)*sinh(x)^6 - 20*cosh(x)^6 + 24*(33*cosh(x)^7 - 63*cosh(x)^5 + 35*cosh(x)^3 -
5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 - 84*cosh(x)^6 + 70*cosh(x)^4 - 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*cosh
(x)^4 + 20*(11*cosh(x)^9 - 36*cosh(x)^7 + 42*cosh(x)^5 - 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^1
0 - 45*cosh(x)^8 + 70*cosh(x)^6 - 50*cosh(x)^4 + 15*cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 12*(cosh(x)^11 -
5*cosh(x)^9 + 10*cosh(x)^7 - 10*cosh(x)^5 + 5*cosh(x)^3 - cosh(x))*sinh(x) + 1)

Sympy [F]

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

[In]

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

[Out]

Integral((a*csch(x)**3)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*csch(x)^3)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*csch(x)^3)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sinh(x)^3)^(5/2), x)