\(\int (a \cosh (x))^{7/2} \, dx\) [15]

Optimal result

Integrand size = 8, antiderivative size = 65 \[ \int (a \cosh (x))^{7/2} \, dx=-\frac {10 i a^4 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{21 \sqrt {a \cosh (x)}}+\frac {10}{21} a^3 \sqrt {a \cosh (x)} \sinh (x)+\frac {2}{7} a (a \cosh (x))^{5/2} \sinh (x) \] Output:

-10/21*I*a^4*cosh(x)^(1/2)*InverseJacobiAM(1/2*I*x,2^(1/2))/(a*cosh(x))^(1 
/2)+10/21*a^3*(a*cosh(x))^(1/2)*sinh(x)+2/7*a*(a*cosh(x))^(5/2)*sinh(x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82 \[ \int (a \cosh (x))^{7/2} \, dx=\frac {a^3 \sqrt {a \cosh (x)} \left (-20 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\sqrt {\cosh (x)} (23 \sinh (x)+3 \sinh (3 x))\right )}{42 \sqrt {\cosh (x)}} \] Input:

Integrate[(a*Cosh[x])^(7/2),x]
 

Output:

(a^3*Sqrt[a*Cosh[x]]*((-20*I)*EllipticF[(I/2)*x, 2] + Sqrt[Cosh[x]]*(23*Si 
nh[x] + 3*Sinh[3*x])))/(42*Sqrt[Cosh[x]])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3115, 3042, 3115, 3042, 3121, 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.

\(\displaystyle \int (a \cosh (x))^{7/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sin \left (\frac {\pi }{2}+i x\right )\right )^{7/2}dx\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {5}{7} a^2 \int (a \cosh (x))^{3/2}dx+\frac {2}{7} a \sinh (x) (a \cosh (x))^{5/2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

Int[(a*Cosh[x])^(7/2),x]
 

Output:

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

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 3121

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(49)=98\).

Time = 5.85 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.23

method	result	size
default	\(\frac {\sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) a \sinh \left (\frac {x}{2}\right )^{2}}\, a^{4} \left (96 \cosh \left (\frac {x}{2}\right )^{9}-240 \cosh \left (\frac {x}{2}\right )^{7}+256 \cosh \left (\frac {x}{2}\right )^{5}+5 \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-144 \cosh \left (\frac {x}{2}\right )^{3}+32 \cosh \left (\frac {x}{2}\right )\right )}{21 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) a}}\)	\(145\)

Input:

int((cosh(x)*a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/21*((2*cosh(1/2*x)^2-1)*a*sinh(1/2*x)^2)^(1/2)*a^4*(96*cosh(1/2*x)^9-240 
*cosh(1/2*x)^7+256*cosh(1/2*x)^5+5*2^(1/2)*(-2*cosh(1/2*x)^2+1)^(1/2)*(-si 
nh(1/2*x)^2)^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-144*cosh(1/2 
*x)^3+32*cosh(1/2*x))/(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)/sinh(1/2*x 
)/((2*cosh(1/2*x)^2-1)*a)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (47) = 94\).

Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.80 \[ \int (a \cosh (x))^{7/2} \, dx=\frac {80 \, \sqrt {\frac {1}{2}} {\left (a^{3} \cosh \left (x\right )^{3} + 3 \, a^{3} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{2} + a^{3} \sinh \left (x\right )^{3}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (3 \, a^{3} \cosh \left (x\right )^{6} + 18 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, a^{3} \sinh \left (x\right )^{6} + 23 \, a^{3} \cosh \left (x\right )^{4} - 23 \, a^{3} \cosh \left (x\right )^{2} + {\left (45 \, a^{3} \cosh \left (x\right )^{2} + 23 \, a^{3}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (15 \, a^{3} \cosh \left (x\right )^{3} + 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \, a^{3} + {\left (45 \, a^{3} \cosh \left (x\right )^{4} + 138 \, a^{3} \cosh \left (x\right )^{2} - 23 \, a^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (9 \, a^{3} \cosh \left (x\right )^{5} + 46 \, a^{3} \cosh \left (x\right )^{3} - 23 \, a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{84 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \] Input:

integrate((a*cosh(x))^(7/2),x, algorithm="fricas")
 

Output:

1/84*(80*sqrt(1/2)*(a^3*cosh(x)^3 + 3*a^3*cosh(x)^2*sinh(x) + 3*a^3*cosh(x 
)*sinh(x)^2 + a^3*sinh(x)^3)*sqrt(a)*weierstrassPInverse(-4, 0, cosh(x) + 
sinh(x)) + (3*a^3*cosh(x)^6 + 18*a^3*cosh(x)*sinh(x)^5 + 3*a^3*sinh(x)^6 + 
 23*a^3*cosh(x)^4 - 23*a^3*cosh(x)^2 + (45*a^3*cosh(x)^2 + 23*a^3)*sinh(x) 
^4 + 4*(15*a^3*cosh(x)^3 + 23*a^3*cosh(x))*sinh(x)^3 - 3*a^3 + (45*a^3*cos 
h(x)^4 + 138*a^3*cosh(x)^2 - 23*a^3)*sinh(x)^2 + 2*(9*a^3*cosh(x)^5 + 46*a 
^3*cosh(x)^3 - 23*a^3*cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(cosh(x)^3 + 3*co 
sh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)
 

Sympy [F(-1)]

Timed out. \[ \int (a \cosh (x))^{7/2} \, dx=\text {Timed out} \] Input:

integrate((a*cosh(x))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*cosh(x))^(7/2),x, algorithm="maxima")
 

Output:

integrate((a*cosh(x))^(7/2), x)
 

Giac [F]

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

integrate((a*cosh(x))^(7/2),x, algorithm="giac")
 

Output:

integrate((a*cosh(x))^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a \cosh (x))^{7/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )\right )}^{7/2} \,d x \] Input:

int((a*cosh(x))^(7/2),x)
 

Output:

int((a*cosh(x))^(7/2), x)
 

Reduce [F]

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

int((a*cosh(x))^(7/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(a)*int(sqrt(cosh(x))*cosh(x)**3,x)*a**3