\(\int \frac {1}{(a \cosh ^3(x))^{5/2}} \, dx\) [133]

Optimal result

Integrand size = 10, antiderivative size = 121 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\frac {154 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}} \] Output:

154/195*I*cosh(x)^(3/2)*EllipticE(I*sinh(1/2*x),2^(1/2))/a^2/(a*cosh(x)^3) 
^(1/2)+154/195*cosh(x)*sinh(x)/a^2/(a*cosh(x)^3)^(1/2)+154/585*tanh(x)/a^2 
/(a*cosh(x)^3)^(1/2)+22/117*sech(x)^2*tanh(x)/a^2/(a*cosh(x)^3)^(1/2)+2/13 
*sech(x)^4*tanh(x)/a^2/(a*cosh(x)^3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\frac {462 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+462 \cosh (x) \sinh (x)+2 \left (77+55 \text {sech}^2(x)+45 \text {sech}^4(x)\right ) \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}} \] Input:

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

Output:

((462*I)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2] + 462*Cosh[x]*Sinh[x] + 2*(77 
 + 55*Sech[x]^2 + 45*Sech[x]^4)*Tanh[x])/(585*a^2*Sqrt[a*Cosh[x]^3])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79, 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, 3116, 3042, 3116, 3042, 3116, 3042, 3116, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3686

\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \cosh ^3(x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{15/2}}dx}{a^2 \sqrt {a \cosh ^3(x)}}\)

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {3}{5} \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}+2 i E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )\right )\right )}{a^2 \sqrt {a \cosh ^3(x)}}\)

Input:

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

Output:

(Cosh[x]^(3/2)*((2*Sinh[x])/(13*Cosh[x]^(13/2)) + (11*((2*Sinh[x])/(9*Cosh 
[x]^(9/2)) + (7*((2*Sinh[x])/(5*Cosh[x]^(5/2)) + (3*((2*I)*EllipticE[(I/2) 
*x, 2] + (2*Sinh[x])/Sqrt[Cosh[x]]))/5))/9))/13))/(a^2*Sqrt[a*Cosh[x]^3])
 

Defintions of rubi rules used

rule 3042

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

rule 3116

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

\[\int \frac {1}{\left (a \cosh \left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]

Input:

int(1/(a*cosh(x)^3)^(5/2),x)
 

Output:

int(1/(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. 1473 vs. \(2 (98) = 196\).

Time = 0.15 (sec) , antiderivative size = 1473, normalized size of antiderivative = 12.17 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="fricas")
 

Output:

4/585*(231*sqrt(1/2)*(cosh(x)^14 + 14*cosh(x)*sinh(x)^13 + sinh(x)^14 + 7* 
(13*cosh(x)^2 + 1)*sinh(x)^12 + 7*cosh(x)^12 + 28*(13*cosh(x)^3 + 3*cosh(x 
))*sinh(x)^11 + 7*(143*cosh(x)^4 + 66*cosh(x)^2 + 3)*sinh(x)^10 + 21*cosh( 
x)^10 + 14*(143*cosh(x)^5 + 110*cosh(x)^3 + 15*cosh(x))*sinh(x)^9 + 7*(429 
*cosh(x)^6 + 495*cosh(x)^4 + 135*cosh(x)^2 + 5)*sinh(x)^8 + 35*cosh(x)^8 + 
 8*(429*cosh(x)^7 + 693*cosh(x)^5 + 315*cosh(x)^3 + 35*cosh(x))*sinh(x)^7 
+ 7*(429*cosh(x)^8 + 924*cosh(x)^6 + 630*cosh(x)^4 + 140*cosh(x)^2 + 5)*si 
nh(x)^6 + 35*cosh(x)^6 + 14*(143*cosh(x)^9 + 396*cosh(x)^7 + 378*cosh(x)^5 
 + 140*cosh(x)^3 + 15*cosh(x))*sinh(x)^5 + 7*(143*cosh(x)^10 + 495*cosh(x) 
^8 + 630*cosh(x)^6 + 350*cosh(x)^4 + 75*cosh(x)^2 + 3)*sinh(x)^4 + 21*cosh 
(x)^4 + 28*(13*cosh(x)^11 + 55*cosh(x)^9 + 90*cosh(x)^7 + 70*cosh(x)^5 + 2 
5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 7*(13*cosh(x)^12 + 66*cosh(x)^10 + 13 
5*cosh(x)^8 + 140*cosh(x)^6 + 75*cosh(x)^4 + 18*cosh(x)^2 + 1)*sinh(x)^2 + 
 7*cosh(x)^2 + 14*(cosh(x)^13 + 6*cosh(x)^11 + 15*cosh(x)^9 + 20*cosh(x)^7 
 + 15*cosh(x)^5 + 6*cosh(x)^3 + cosh(x))*sinh(x) + 1)*sqrt(a)*weierstrassZ 
eta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + (231*cosh(x)^1 
4 + 3234*cosh(x)*sinh(x)^13 + 231*sinh(x)^14 + 77*(273*cosh(x)^2 + 20)*sin 
h(x)^12 + 1540*cosh(x)^12 + 924*(91*cosh(x)^3 + 20*cosh(x))*sinh(x)^11 + 1 
1*(21021*cosh(x)^4 + 9240*cosh(x)^2 + 397)*sinh(x)^10 + 4367*cosh(x)^10 + 
22*(21021*cosh(x)^5 + 15400*cosh(x)^3 + 1985*cosh(x))*sinh(x)^9 + (6936...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a*cosh(x)**3)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cosh}\left (x\right )}^3\right )}^{5/2}} \,d x \] Input:

int(1/(a*cosh(x)^3)^(5/2),x)
 

Output:

int(1/(a*cosh(x)^3)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cosh \left (x \right )}}{\cosh \left (x \right )^{8}}d x \right )}{a^{3}} \] Input:

int(1/(a*cosh(x)^3)^(5/2),x)
 

Output:

(sqrt(a)*int(sqrt(cosh(x))/cosh(x)**8,x))/a**3