\(\int (a \cosh ^3(x))^{3/2} \, dx\) [129]

Optimal result

Integrand size = 10, antiderivative size = 71 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=-\frac {14 i a \sqrt {a \cosh ^3(x)} E\left (\left .\frac {i x}{2}\right |2\right )}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x) \]

[Out]

-14/15*I*a*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*cosh(x)^3)^(1/2)/cosh(x)^(3/2
)+14/45*a*sinh(x)*(a*cosh(x)^3)^(1/2)+2/9*a*cosh(x)^2*sinh(x)*(a*cosh(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2719} \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\frac {14}{45} a \sinh (x) \sqrt {a \cosh ^3(x)}-\frac {14 i a E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh ^3(x)}}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {2}{9} a \sinh (x) \cosh ^2(x) \sqrt {a \cosh ^3(x)} \]

[In]

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

[Out]

(((-14*I)/15)*a*Sqrt[a*Cosh[x]^3]*EllipticE[(I/2)*x, 2])/Cosh[x]^(3/2) + (14*a*Sqrt[a*Cosh[x]^3]*Sinh[x])/45 +
(2*a*Cosh[x]^2*Sqrt[a*Cosh[x]^3]*Sinh[x])/9

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 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 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 \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {9}{2}}(x) \, dx}{\cosh ^{\frac {3}{2}}(x)} \\ & = \frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (7 a \sqrt {a \cosh ^3(x)}\right ) \int \cosh ^{\frac {5}{2}}(x) \, dx}{9 \cosh ^{\frac {3}{2}}(x)} \\ & = \frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {\left (7 a \sqrt {a \cosh ^3(x)}\right ) \int \sqrt {\cosh (x)} \, dx}{15 \cosh ^{\frac {3}{2}}(x)} \\ & = -\frac {14 i a \sqrt {a \cosh ^3(x)} E\left (\left .\frac {i x}{2}\right |2\right )}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cosh ^3(x)} \sinh (x)+\frac {2}{9} a \cosh ^2(x) \sqrt {a \cosh ^3(x)} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\frac {\left (a \cosh ^3(x)\right )^{3/2} \left (-168 i E\left (\left .\frac {i x}{2}\right |2\right )+\sqrt {\cosh (x)} (38 \sinh (2 x)+5 \sinh (4 x))\right )}{180 \cosh ^{\frac {9}{2}}(x)} \]

[In]

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

[Out]

((a*Cosh[x]^3)^(3/2)*((-168*I)*EllipticE[(I/2)*x, 2] + Sqrt[Cosh[x]]*(38*Sinh[2*x] + 5*Sinh[4*x])))/(180*Cosh[
x]^(9/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 317, normalized size of antiderivative = 4.46 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=-\frac {336 \, {\left (\sqrt {2} a \cosh \left (x\right )^{4} + 4 \, \sqrt {2} a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \sqrt {2} a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} a \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} + 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} + 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} + 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} + 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} + 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} + 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} - 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \cosh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]

[In]

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

[Out]

-1/360*(336*(sqrt(2)*a*cosh(x)^4 + 4*sqrt(2)*a*cosh(x)^3*sinh(x) + 6*sqrt(2)*a*cosh(x)^2*sinh(x)^2 + 4*sqrt(2)
*a*cosh(x)*sinh(x)^3 + sqrt(2)*a*sinh(x)^4)*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x)
+ sinh(x))) - (5*a*cosh(x)^8 + 40*a*cosh(x)*sinh(x)^7 + 5*a*sinh(x)^8 + 38*a*cosh(x)^6 + 2*(70*a*cosh(x)^2 + 1
9*a)*sinh(x)^6 + 4*(70*a*cosh(x)^3 + 57*a*cosh(x))*sinh(x)^5 - 336*a*cosh(x)^4 + 2*(175*a*cosh(x)^4 + 285*a*co
sh(x)^2 - 168*a)*sinh(x)^4 + 8*(35*a*cosh(x)^5 + 95*a*cosh(x)^3 - 168*a*cosh(x))*sinh(x)^3 - 38*a*cosh(x)^2 +
2*(70*a*cosh(x)^6 + 285*a*cosh(x)^4 - 1008*a*cosh(x)^2 - 19*a)*sinh(x)^2 + 4*(10*a*cosh(x)^7 + 57*a*cosh(x)^5
- 336*a*cosh(x)^3 - 19*a*cosh(x))*sinh(x) - 5*a)*sqrt(a*cosh(x)))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)
^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x)^3)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cosh(x)^3)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a*cosh(x)^3)^(3/2), x)