\(\int (a \cosh ^3(x))^{5/2} \, dx\) [128]

Optimal result

Integrand size = 10, antiderivative size = 121 \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=-\frac {26 i a^2 \sqrt {a \cosh ^3(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{77 \cosh ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cosh (x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{165} a^2 \cosh ^3(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{15} a^2 \cosh ^5(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {26}{77} a^2 \sqrt {a \cosh ^3(x)} \tanh (x) \]

[Out]

-26/77*I*a^2*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))*(a*cosh(x)^3)^(1/2)/cosh(x)^(3
/2)+78/385*a^2*cosh(x)*sinh(x)*(a*cosh(x)^3)^(1/2)+26/165*a^2*cosh(x)^3*sinh(x)*(a*cosh(x)^3)^(1/2)+2/15*a^2*c
osh(x)^5*sinh(x)*(a*cosh(x)^3)^(1/2)+26/77*a^2*(a*cosh(x)^3)^(1/2)*tanh(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2720} \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\frac {26}{165} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^3(x)}+\frac {78}{385} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^3(x)}+\frac {26}{77} a^2 \tanh (x) \sqrt {a \cosh ^3(x)}-\frac {26 i a^2 \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \cosh ^3(x)}}{77 \cosh ^{\frac {3}{2}}(x)}+\frac {2}{15} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^3(x)} \]

[In]

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

[Out]

(((-26*I)/77)*a^2*Sqrt[a*Cosh[x]^3]*EllipticF[(I/2)*x, 2])/Cosh[x]^(3/2) + (78*a^2*Cosh[x]*Sqrt[a*Cosh[x]^3]*S
inh[x])/385 + (26*a^2*Cosh[x]^3*Sqrt[a*Cosh[x]^3]*Sinh[x])/165 + (2*a^2*Cosh[x]^5*Sqrt[a*Cosh[x]^3]*Sinh[x])/1
5 + (26*a^2*Sqrt[a*Cosh[x]^3]*Tanh[x])/77

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.54 \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\frac {a \left (a \cosh ^3(x)\right )^{3/2} \left (-12480 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\sqrt {\cosh (x)} (15465 \sinh (x)+3657 \sinh (3 x)+749 \sinh (5 x)+77 \sinh (7 x))\right )}{36960 \cosh ^{\frac {9}{2}}(x)} \]

[In]

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

[Out]

(a*(a*Cosh[x]^3)^(3/2)*((-12480*I)*EllipticF[(I/2)*x, 2] + Sqrt[Cosh[x]]*(15465*Sinh[x] + 3657*Sinh[3*x] + 749
*Sinh[5*x] + 77*Sinh[7*x])))/(36960*Cosh[x]^(9/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 823, normalized size of antiderivative = 6.80 \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((a*cosh(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*c
osh(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 + 5485
5*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 +
18285*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)^1
0 + 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*cosh(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*co
sh(x)*sinh(x)^6 + sinh(x)^7)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x)^3)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cosh(x)^3)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a*cosh(x)^3)^(5/2), x)