\(\int (a \sinh ^3(x))^{5/2} \, dx\) [146]

Optimal result

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

[Out]

-26/77*a^2*coth(x)*(a*sinh(x)^3)^(1/2)+78/385*a^2*cosh(x)*sinh(x)*(a*sinh(x)^3)^(1/2)-26/165*a^2*cosh(x)*sinh(
x)^3*(a*sinh(x)^3)^(1/2)+2/15*a^2*cosh(x)*sinh(x)^5*(a*sinh(x)^3)^(1/2)+26/77*I*a^2*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*sinh(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.04 (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, 2715, 2721, 2720} \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=-\frac {26}{165} a^2 \sinh ^3(x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {78}{385} a^2 \sinh (x) \cosh (x) \sqrt {a \sinh ^3(x)}+\frac {2}{15} a^2 \sinh ^5(x) \cosh (x) \sqrt {a \sinh ^3(x)}-\frac {26}{77} a^2 \coth (x) \sqrt {a \sinh ^3(x)}+\frac {26}{77} i a^2 \sqrt {i \sinh (x)} \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {a \sinh ^3(x)} \]

[In]

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

[Out]

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

Rule 2715

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.50 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\frac {a^2 \text {csch}(x) \left (-15465 \cosh (x)+3657 \cosh (3 x)-749 \cosh (5 x)+77 \cosh (7 x)-\frac {12480 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right )}{\sqrt {i \sinh (x)}}\right ) \sqrt {a \sinh ^3(x)}}{36960} \]

[In]

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

[Out]

(a^2*Csch[x]*(-15465*Cosh[x] + 3657*Cosh[3*x] - 749*Cosh[5*x] + 77*Cosh[7*x] - (12480*EllipticF[(Pi - (2*I)*x)
/4, 2])/Sqrt[I*Sinh[x]])*Sqrt[a*Sinh[x]^3])/36960

Maple [F]

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

[In]

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

[Out]

int((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 = 0.11 (sec) , antiderivative size = 823, normalized size of antiderivative = 6.10 \[ \int \left (a \sinh ^3(x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/73920*(24960*(sqrt(2)*a^2*cosh(x)^7 + 7*sqrt(2)*a^2*cosh(x)^6*sinh(x) + 21*sqrt(2)*a^2*cosh(x)^5*sinh(x)^2 +
35*sqrt(2)*a^2*cosh(x)^4*sinh(x)^3 + 35*sqrt(2)*a^2*cosh(x)^3*sinh(x)^4 + 21*sqrt(2)*a^2*cosh(x)^2*sinh(x)^5
+ 7*sqrt(2)*a^2*cosh(x)*sinh(x)^6 + sqrt(2)*a^2*sinh(x)^7)*sqrt(a)*weierstrassPInverse(4, 0, cosh(x) + sinh(x)
) + (77*a^2*cosh(x)^14 + 1078*a^2*cosh(x)*sinh(x)^13 + 77*a^2*sinh(x)^14 - 749*a^2*cosh(x)^12 + 7*(1001*a^2*co
sh(x)^2 - 107*a^2)*sinh(x)^12 + 3657*a^2*cosh(x)^10 + 28*(1001*a^2*cosh(x)^3 - 321*a^2*cosh(x))*sinh(x)^11 + (
77077*a^2*cosh(x)^4 - 49434*a^2*cosh(x)^2 + 3657*a^2)*sinh(x)^10 - 15465*a^2*cosh(x)^8 + 2*(77077*a^2*cosh(x)^
5 - 82390*a^2*cosh(x)^3 + 18285*a^2*cosh(x))*sinh(x)^9 + 3*(77077*a^2*cosh(x)^6 - 123585*a^2*cosh(x)^4 + 54855
*a^2*cosh(x)^2 - 5155*a^2)*sinh(x)^8 - 15465*a^2*cosh(x)^6 + 24*(11011*a^2*cosh(x)^7 - 24717*a^2*cosh(x)^5 + 1
8285*a^2*cosh(x)^3 - 5155*a^2*cosh(x))*sinh(x)^7 + 3*(77077*a^2*cosh(x)^8 - 230692*a^2*cosh(x)^6 + 255990*a^2*
cosh(x)^4 - 144340*a^2*cosh(x)^2 - 5155*a^2)*sinh(x)^6 + 3657*a^2*cosh(x)^4 + 2*(77077*a^2*cosh(x)^9 - 296604*
a^2*cosh(x)^7 + 460782*a^2*cosh(x)^5 - 433020*a^2*cosh(x)^3 - 46395*a^2*cosh(x))*sinh(x)^5 + (77077*a^2*cosh(x
)^10 - 370755*a^2*cosh(x)^8 + 767970*a^2*cosh(x)^6 - 1082550*a^2*cosh(x)^4 - 231975*a^2*cosh(x)^2 + 3657*a^2)*
sinh(x)^4 - 749*a^2*cosh(x)^2 + 4*(7007*a^2*cosh(x)^11 - 41195*a^2*cosh(x)^9 + 109710*a^2*cosh(x)^7 - 216510*a
^2*cosh(x)^5 - 77325*a^2*cosh(x)^3 + 3657*a^2*cosh(x))*sinh(x)^3 + (7007*a^2*cosh(x)^12 - 49434*a^2*cosh(x)^10
+ 164565*a^2*cosh(x)^8 - 433020*a^2*cosh(x)^6 - 231975*a^2*cosh(x)^4 + 21942*a^2*cosh(x)^2 - 749*a^2)*sinh(x)
^2 + 77*a^2 + 2*(539*a^2*cosh(x)^13 - 4494*a^2*cosh(x)^11 + 18285*a^2*cosh(x)^9 - 61860*a^2*cosh(x)^7 - 46395*
a^2*cosh(x)^5 + 7314*a^2*cosh(x)^3 - 749*a^2*cosh(x))*sinh(x))*sqrt(a*sinh(x)))/(cosh(x)^7 + 7*cosh(x)^6*sinh(
x) + 21*cosh(x)^5*sinh(x)^2 + 35*cosh(x)^4*sinh(x)^3 + 35*cosh(x)^3*sinh(x)^4 + 21*cosh(x)^2*sinh(x)^5 + 7*cos
h(x)*sinh(x)^6 + sinh(x)^7)

Sympy [F]

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

[In]

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

[Out]

Integral((a*sinh(x)**3)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*sinh(x)^3)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*sinh(x)^3)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a*sinh(x)^3)^(5/2), x)