\(\int \frac {1}{(a \sinh ^3(x))^{5/2}} \, dx\) [151]

Optimal result

Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]

[Out]

-154/585*coth(x)/a^2/(a*sinh(x)^3)^(1/2)+22/117*coth(x)*csch(x)^2/a^2/(a*sinh(x)^3)^(1/2)-2/13*coth(x)*csch(x)
^4/a^2/(a*sinh(x)^3)^(1/2)+154/195*cosh(x)*sinh(x)/a^2/(a*sinh(x)^3)^(1/2)-154/195*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))*sinh(x)^2/a^2/(I*sinh(x))^(1/2)/(a*sinh(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 = {3286, 2716, 2721, 2719} \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\frac {154 \sinh (x) \cosh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}} \]

[In]

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

[Out]

(-154*Coth[x])/(585*a^2*Sqrt[a*Sinh[x]^3]) + (22*Coth[x]*Csch[x]^2)/(117*a^2*Sqrt[a*Sinh[x]^3]) - (2*Coth[x]*C
sch[x]^4)/(13*a^2*Sqrt[a*Sinh[x]^3]) + (154*Cosh[x]*Sinh[x])/(195*a^2*Sqrt[a*Sinh[x]^3]) - (((154*I)/195)*Elli
pticE[Pi/4 - (I/2)*x, 2]*Sinh[x]^2)/(a^2*Sqrt[I*Sinh[x]]*Sqrt[a*Sinh[x]^3])

Rule 2716

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

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 2721

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

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \sinh ^3(x)}} \\ & = -\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (11 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \sinh ^3(x)}} \\ & = \frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \sinh ^3(x)}} \\ & = -\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \sinh ^3(x)}} \\ & = -\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sinh (x)} \, dx}{195 a^2 \sqrt {a \sinh ^3(x)}} \\ & = -\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^2(x)\right ) \int \sqrt {i \sinh (x)} \, dx}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \\ & = -\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\frac {-2 \coth (x) \left (77-55 \text {csch}^2(x)+45 \text {csch}^4(x)\right )+462 i E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) (i \sinh (x))^{3/2}+462 \cosh (x) \sinh (x)}{585 a^2 \sqrt {a \sinh ^3(x)}} \]

[In]

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

[Out]

(-2*Coth[x]*(77 - 55*Csch[x]^2 + 45*Csch[x]^4) + (462*I)*EllipticE[(Pi - (2*I)*x)/4, 2]*(I*Sinh[x])^(3/2) + 46
2*Cosh[x]*Sinh[x])/(585*a^2*Sqrt[a*Sinh[x]^3])

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 1676, normalized size of antiderivative = 12.41 \[ \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/585*(231*(sqrt(2)*cosh(x)^14 + 14*sqrt(2)*cosh(x)*sinh(x)^13 + sqrt(2)*sinh(x)^14 + 7*(13*sqrt(2)*cosh(x)^2
- sqrt(2))*sinh(x)^12 - 7*sqrt(2)*cosh(x)^12 + 28*(13*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*sinh(x)^11 + 7*(1
43*sqrt(2)*cosh(x)^4 - 66*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^10 + 21*sqrt(2)*cosh(x)^10 + 14*(143*sqrt(2)*
cosh(x)^5 - 110*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x))*sinh(x)^9 + 7*(429*sqrt(2)*cosh(x)^6 - 495*sqrt(2)*cos
h(x)^4 + 135*sqrt(2)*cosh(x)^2 - 5*sqrt(2))*sinh(x)^8 - 35*sqrt(2)*cosh(x)^8 + 8*(429*sqrt(2)*cosh(x)^7 - 693*
sqrt(2)*cosh(x)^5 + 315*sqrt(2)*cosh(x)^3 - 35*sqrt(2)*cosh(x))*sinh(x)^7 + 7*(429*sqrt(2)*cosh(x)^8 - 924*sqr
t(2)*cosh(x)^6 + 630*sqrt(2)*cosh(x)^4 - 140*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh(x)^6 + 35*sqrt(2)*cosh(x)^6 +
14*(143*sqrt(2)*cosh(x)^9 - 396*sqrt(2)*cosh(x)^7 + 378*sqrt(2)*cosh(x)^5 - 140*sqrt(2)*cosh(x)^3 + 15*sqrt(2
)*cosh(x))*sinh(x)^5 + 7*(143*sqrt(2)*cosh(x)^10 - 495*sqrt(2)*cosh(x)^8 + 630*sqrt(2)*cosh(x)^6 - 350*sqrt(2)
*cosh(x)^4 + 75*sqrt(2)*cosh(x)^2 - 3*sqrt(2))*sinh(x)^4 - 21*sqrt(2)*cosh(x)^4 + 28*(13*sqrt(2)*cosh(x)^11 -
55*sqrt(2)*cosh(x)^9 + 90*sqrt(2)*cosh(x)^7 - 70*sqrt(2)*cosh(x)^5 + 25*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))
*sinh(x)^3 + 7*(13*sqrt(2)*cosh(x)^12 - 66*sqrt(2)*cosh(x)^10 + 135*sqrt(2)*cosh(x)^8 - 140*sqrt(2)*cosh(x)^6
+ 75*sqrt(2)*cosh(x)^4 - 18*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 7*sqrt(2)*cosh(x)^2 + 14*(sqrt(2)*cosh(x)
^13 - 6*sqrt(2)*cosh(x)^11 + 15*sqrt(2)*cosh(x)^9 - 20*sqrt(2)*cosh(x)^7 + 15*sqrt(2)*cosh(x)^5 - 6*sqrt(2)*co
sh(x)^3 + sqrt(2)*cosh(x))*sinh(x) - sqrt(2))*sqrt(a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(x)
+ sinh(x))) + 2*(231*cosh(x)^14 + 3234*cosh(x)*sinh(x)^13 + 231*sinh(x)^14 + 77*(273*cosh(x)^2 - 20)*sinh(x)^1
2 - 1540*cosh(x)^12 + 924*(91*cosh(x)^3 - 20*cosh(x))*sinh(x)^11 + 11*(21021*cosh(x)^4 - 9240*cosh(x)^2 + 397)
*sinh(x)^10 + 4367*cosh(x)^10 + 22*(21021*cosh(x)^5 - 15400*cosh(x)^3 + 1985*cosh(x))*sinh(x)^9 + (693693*cosh
(x)^6 - 762300*cosh(x)^4 + 196515*cosh(x)^2 - 6808)*sinh(x)^8 - 6808*cosh(x)^8 + 8*(99099*cosh(x)^7 - 152460*c
osh(x)^5 + 65505*cosh(x)^3 - 6808*cosh(x))*sinh(x)^7 + (693693*cosh(x)^8 - 1422960*cosh(x)^6 + 917070*cosh(x)^
4 - 190624*cosh(x)^2 + 1277)*sinh(x)^6 + 1277*cosh(x)^6 + 2*(231231*cosh(x)^9 - 609840*cosh(x)^7 + 550242*cosh
(x)^5 - 190624*cosh(x)^3 + 3831*cosh(x))*sinh(x)^5 + (231231*cosh(x)^10 - 762300*cosh(x)^8 + 917070*cosh(x)^6
- 476560*cosh(x)^4 + 19155*cosh(x)^2 - 484)*sinh(x)^4 - 484*cosh(x)^4 + 4*(21021*cosh(x)^11 - 84700*cosh(x)^9
+ 131010*cosh(x)^7 - 95312*cosh(x)^5 + 6385*cosh(x)^3 - 484*cosh(x))*sinh(x)^3 + (21021*cosh(x)^12 - 101640*co
sh(x)^10 + 196515*cosh(x)^8 - 190624*cosh(x)^6 + 19155*cosh(x)^4 - 2904*cosh(x)^2 + 77)*sinh(x)^2 + 77*cosh(x)
^2 + 2*(1617*cosh(x)^13 - 9240*cosh(x)^11 + 21835*cosh(x)^9 - 27232*cosh(x)^7 + 3831*cosh(x)^5 - 968*cosh(x)^3
+ 77*cosh(x))*sinh(x))*sqrt(a*sinh(x)))/(a^3*cosh(x)^14 + 14*a^3*cosh(x)*sinh(x)^13 + a^3*sinh(x)^14 - 7*a^3*
cosh(x)^12 + 21*a^3*cosh(x)^10 + 7*(13*a^3*cosh(x)^2 - a^3)*sinh(x)^12 + 28*(13*a^3*cosh(x)^3 - 3*a^3*cosh(x))
*sinh(x)^11 - 35*a^3*cosh(x)^8 + 7*(143*a^3*cosh(x)^4 - 66*a^3*cosh(x)^2 + 3*a^3)*sinh(x)^10 + 14*(143*a^3*cos
h(x)^5 - 110*a^3*cosh(x)^3 + 15*a^3*cosh(x))*sinh(x)^9 + 35*a^3*cosh(x)^6 + 7*(429*a^3*cosh(x)^6 - 495*a^3*cos
h(x)^4 + 135*a^3*cosh(x)^2 - 5*a^3)*sinh(x)^8 + 8*(429*a^3*cosh(x)^7 - 693*a^3*cosh(x)^5 + 315*a^3*cosh(x)^3 -
35*a^3*cosh(x))*sinh(x)^7 - 21*a^3*cosh(x)^4 + 7*(429*a^3*cosh(x)^8 - 924*a^3*cosh(x)^6 + 630*a^3*cosh(x)^4 -
140*a^3*cosh(x)^2 + 5*a^3)*sinh(x)^6 + 14*(143*a^3*cosh(x)^9 - 396*a^3*cosh(x)^7 + 378*a^3*cosh(x)^5 - 140*a^
3*cosh(x)^3 + 15*a^3*cosh(x))*sinh(x)^5 + 7*a^3*cosh(x)^2 + 7*(143*a^3*cosh(x)^10 - 495*a^3*cosh(x)^8 + 630*a^
3*cosh(x)^6 - 350*a^3*cosh(x)^4 + 75*a^3*cosh(x)^2 - 3*a^3)*sinh(x)^4 + 28*(13*a^3*cosh(x)^11 - 55*a^3*cosh(x)
^9 + 90*a^3*cosh(x)^7 - 70*a^3*cosh(x)^5 + 25*a^3*cosh(x)^3 - 3*a^3*cosh(x))*sinh(x)^3 - a^3 + 7*(13*a^3*cosh(
x)^12 - 66*a^3*cosh(x)^10 + 135*a^3*cosh(x)^8 - 140*a^3*cosh(x)^6 + 75*a^3*cosh(x)^4 - 18*a^3*cosh(x)^2 + a^3)
*sinh(x)^2 + 14*(a^3*cosh(x)^13 - 6*a^3*cosh(x)^11 + 15*a^3*cosh(x)^9 - 20*a^3*cosh(x)^7 + 15*a^3*cosh(x)^5 -
6*a^3*cosh(x)^3 + a^3*cosh(x))*sinh(x))

Sympy [F]

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

[In]

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

[Out]

Integral((a*sinh(x)**3)**(-5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*sinh(x)^3)^(-5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*sinh(x)^3)^(-5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a*sinh(x)^3)^(5/2), x)