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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\frac {\left (a \cosh ^3(x)\right )^{3/2} \left (-168 i E\left (\left .\frac {i x}{2}\right |2\right )+\sqrt {\cosh (x)} (38 \sinh (2 x)+5 \sinh (4 x))\right )}{180 \cosh ^{\frac {9}{2}}(x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3119}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a \cosh ^3(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )^3\right )^{3/2}dx\)

\(\Big \downarrow \) 3686

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \int \cosh ^{\frac {9}{2}}(x)dx}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \int \sin \left (i x+\frac {\pi }{2}\right )^{9/2}dx}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \left (\frac {7}{9} \int \cosh ^{\frac {5}{2}}(x)dx+\frac {2}{9} \sinh (x) \cosh ^{\frac {7}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \left (\frac {2}{9} \sinh (x) \cosh ^{\frac {7}{2}}(x)+\frac {7}{9} \int \sin \left (i x+\frac {\pi }{2}\right )^{5/2}dx\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\cosh (x)}dx+\frac {2}{5} \sinh (x) \cosh ^{\frac {3}{2}}(x)\right )+\frac {2}{9} \sinh (x) \cosh ^{\frac {7}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \left (\frac {2}{9} \sinh (x) \cosh ^{\frac {7}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \cosh ^{\frac {3}{2}}(x)+\frac {3}{5} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a \sqrt {a \cosh ^3(x)} \left (\frac {2}{9} \sinh (x) \cosh ^{\frac {7}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \cosh ^{\frac {3}{2}}(x)-\frac {6}{5} i E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

Input:

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

Output:

(a*Sqrt[a*Cosh[x]^3]*((2*Cosh[x]^(7/2)*Sinh[x])/9 + (7*(((-6*I)/5)*Ellipti 
cE[(I/2)*x, 2] + (2*Cosh[x]^(3/2)*Sinh[x])/5))/9))/Cosh[x]^(3/2)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3119

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 3686

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ 
n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Si 
n[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]])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (56) = 112\).

Time = 0.09 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.30 \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=-\frac {672 \, \sqrt {\frac {1}{2}} {\left (a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left (5 \, a \cosh \left (x\right )^{8} + 40 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + 5 \, a \sinh \left (x\right )^{8} + 38 \, a \cosh \left (x\right )^{6} + 2 \, {\left (70 \, a \cosh \left (x\right )^{2} + 19 \, a\right )} \sinh \left (x\right )^{6} + 4 \, {\left (70 \, a \cosh \left (x\right )^{3} + 57 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{4} + 2 \, {\left (175 \, a \cosh \left (x\right )^{4} + 285 \, a \cosh \left (x\right )^{2} - 168 \, a\right )} \sinh \left (x\right )^{4} + 8 \, {\left (35 \, a \cosh \left (x\right )^{5} + 95 \, a \cosh \left (x\right )^{3} - 168 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 38 \, a \cosh \left (x\right )^{2} + 2 \, {\left (70 \, a \cosh \left (x\right )^{6} + 285 \, a \cosh \left (x\right )^{4} - 1008 \, a \cosh \left (x\right )^{2} - 19 \, a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (10 \, a \cosh \left (x\right )^{7} + 57 \, a \cosh \left (x\right )^{5} - 336 \, a \cosh \left (x\right )^{3} - 19 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {a \cosh \left (x\right )}}{360 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \] Input:

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

Output:

-1/360*(672*sqrt(1/2)*(a*cosh(x)^4 + 4*a*cosh(x)^3*sinh(x) + 6*a*cosh(x)^2 
*sinh(x)^2 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4)*sqrt(a)*weierstrassZeta( 
-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) - (5*a*cosh(x)^8 + 4 
0*a*cosh(x)*sinh(x)^7 + 5*a*sinh(x)^8 + 38*a*cosh(x)^6 + 2*(70*a*cosh(x)^2 
 + 19*a)*sinh(x)^6 + 4*(70*a*cosh(x)^3 + 57*a*cosh(x))*sinh(x)^5 - 336*a*c 
osh(x)^4 + 2*(175*a*cosh(x)^4 + 285*a*cosh(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))*si 
nh(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. \[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a \cosh ^3(x)\right )^{3/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (x \right )}\, \cosh \left (x \right )^{4}d x \right ) a \] Input:

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

Output:

sqrt(a)*int(sqrt(cosh(x))*cosh(x)**4,x)*a