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

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.54 \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\frac {a \left (a \cosh ^3(x)\right )^{3/2} \left (-12480 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\sqrt {\cosh (x)} (15465 \sinh (x)+3657 \sinh (3 x)+749 \sinh (5 x)+77 \sinh (7 x))\right )}{36960 \cosh ^{\frac {9}{2}}(x)} \] Input:

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

Output:

(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*Co 
sh[x]^(9/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3120}

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.

\(\displaystyle \int \left (a \cosh ^3(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3686

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \int \cosh ^{\frac {15}{2}}(x)dx}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \int \sin \left (i x+\frac {\pi }{2}\right )^{15/2}dx}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {13}{15} \int \cosh ^{\frac {11}{2}}(x)dx+\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)+\frac {13}{15} \int \sin \left (i x+\frac {\pi }{2}\right )^{11/2}dx\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \int \cosh ^{\frac {7}{2}}(x)dx+\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)+\frac {13}{15} \left (\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)+\frac {9}{11} \int \sin \left (i x+\frac {\pi }{2}\right )^{7/2}dx\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \cosh ^{\frac {3}{2}}(x)dx+\frac {2}{7} \sinh (x) \cosh ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)+\frac {13}{15} \left (\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)+\frac {9}{11} \left (\frac {2}{7} \sinh (x) \cosh ^{\frac {5}{2}}(x)+\frac {5}{7} \int \sin \left (i x+\frac {\pi }{2}\right )^{3/2}dx\right )\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cosh (x)}}dx+\frac {2}{3} \sinh (x) \sqrt {\cosh (x)}\right )+\frac {2}{7} \sinh (x) \cosh ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)+\frac {13}{15} \left (\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)+\frac {9}{11} \left (\frac {2}{7} \sinh (x) \cosh ^{\frac {5}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \sqrt {\cosh (x)}+\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {a^2 \sqrt {a \cosh ^3(x)} \left (\frac {2}{15} \sinh (x) \cosh ^{\frac {13}{2}}(x)+\frac {13}{15} \left (\frac {2}{11} \sinh (x) \cosh ^{\frac {9}{2}}(x)+\frac {9}{11} \left (\frac {2}{7} \sinh (x) \cosh ^{\frac {5}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \sqrt {\cosh (x)}-\frac {2}{3} i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )\right )\right )\right )\right )}{\cosh ^{\frac {3}{2}}(x)}\)

Input:

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

Output:

(a^2*Sqrt[a*Cosh[x]^3]*((2*Cosh[x]^(13/2)*Sinh[x])/15 + (13*((2*Cosh[x]^(9 
/2)*Sinh[x])/11 + (9*((2*Cosh[x]^(5/2)*Sinh[x])/7 + (5*(((-2*I)/3)*Ellipti 
cF[(I/2)*x, 2] + (2*Sqrt[Cosh[x]]*Sinh[x])/3))/7))/11))/15))/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 3120

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 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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (95) = 190\).

Time = 0.11 (sec) , antiderivative size = 802, normalized size of antiderivative = 6.63 \[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/73920*(49920*sqrt(1/2)*(a^2*cosh(x)^7 + 7*a^2*cosh(x)^6*sinh(x) + 21*a^2 
*cosh(x)^5*sinh(x)^2 + 35*a^2*cosh(x)^4*sinh(x)^3 + 35*a^2*cosh(x)^3*sinh( 
x)^4 + 21*a^2*cosh(x)^2*sinh(x)^5 + 7*a^2*cosh(x)*sinh(x)^6 + 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*cosh(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*(770 
77*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 + 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*co 
sh(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 - 4 
6395*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 - 3 
657*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*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (x \right )}\, \cosh \left (x \right )^{7}d x \right ) a^{2} \] Input:

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

Output:

sqrt(a)*int(sqrt(cosh(x))*cosh(x)**7,x)*a**2