3.1.84 \(\int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx\) [84]

3.1.84.1 Optimal result

Integrand size = 10, antiderivative size = 177 \[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}} \]

output

-2/3*b*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(3/2)-8/3*a*b*sinh(x)/(a^2-b^2)^2/( 
a+b*cosh(x))^(1/2)-8/3*I*a*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*s 
inh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/(a^2-b^2)^2/((a+b* 
cosh(x))/(a+b))^(1/2)+2/3*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I* 
sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cosh(x))/(a+b))^(1/2)/(a^2-b^2) 
/(a+b*cosh(x))^(1/2)
 

3.1.84.2 Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=\frac {-8 i a (a+b)^2 \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i (a-b) (a+b)^2 \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b \left (-5 a^2+b^2-4 a b \cosh (x)\right ) \sinh (x)}{3 (a-b)^2 (a+b)^2 (a+b \cosh (x))^{3/2}} \]

input

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

output

((-8*I)*a*(a + b)^2*((a + b*Cosh[x])/(a + b))^(3/2)*EllipticE[(I/2)*x, (2* 
b)/(a + b)] + (2*I)*(a - b)*(a + b)^2*((a + b*Cosh[x])/(a + b))^(3/2)*Elli 
pticF[(I/2)*x, (2*b)/(a + b)] + 2*b*(-5*a^2 + b^2 - 4*a*b*Cosh[x])*Sinh[x] 
)/(3*(a - b)^2*(a + b)^2*(a + b*Cosh[x])^(3/2))
 

3.1.84.3 Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 3143, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3143

\(\displaystyle -\frac {2 \int -\frac {3 a-b \cosh (x)}{2 (a+b \cosh (x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 a-b \cosh (x)}{(a+b \cosh (x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {2 \int -\frac {3 a^2+4 b \cosh (x) a+b^2}{2 \sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 a^2+4 b \cosh (x) a+b^2}{\sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\int \frac {3 a^2+4 b \sin \left (i x+\frac {\pi }{2}\right ) a+b^2}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {4 a \int \sqrt {a+b \cosh (x)}dx-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {4 a \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {4 a \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {4 a \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\frac {\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\frac {\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {8 a b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

input

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

output

(-2*b*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) + ((((-8*I)*a*Sqrt[a 
+ b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + 
b)] + ((2*I)*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, 
(2*b)/(a + b)])/Sqrt[a + b*Cosh[x]])/(a^2 - b^2) - (8*a*b*Sinh[x])/((a^2 - 
 b^2)*Sqrt[a + b*Cosh[x]]))/(3*(a^2 - b^2))
 

3.1.84.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3143

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos 
[c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp 
[1/((n + 1)*(a^2 - b^2))   Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) 
- b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
 

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b 
, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 3233

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + 
 f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( 
m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

3.1.84.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(193)=386\).

Time = 2.63 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.59

method	result	size
default	\(\frac {\sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (-\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{3 b \left (a -b \right ) \left (a +b \right ) \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{2}}-\frac {16 \sinh \left (\frac {x}{2}\right )^{2} b \cosh \left (\frac {x}{2}\right ) a}{3 \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {2 \left (3 a -b \right ) \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )}{\left (3 a^{3}+3 a^{2} b -3 a \,b^{2}-3 b^{3}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}-\frac {32 a b \left (-a +b \right ) \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \left (\operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )-\operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 a -2 b \right )}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\)	\(459\)

input

int(1/(a+b*cosh(x))^(5/2),x,method=_RETURNVERBOSE)
 

output

((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(-1/3/b/(a-b)/(a+b)*cosh(1/2 
*x)*(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2*(a-b) 
/b)^2-16/3*sinh(1/2*x)^2*b/(a-b)^2/(a+b)^2*cosh(1/2*x)*a/((2*cosh(1/2*x)^2 
*b+a-b)*sinh(1/2*x)^2)^(1/2)+2*(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3)/(-2*b 
/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2) 
/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b 
/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-32/3*a*b/(a+b)^2/(a-b)^2*(-a+b)/(- 
2*b/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1 
/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/(2*a-2*b)*(EllipticF(cos 
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-EllipticE(cosh(1/2*x 
)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))))/sinh(1/2*x)/(2*sinh(1/2*x 
)^2*b+a+b)^(1/2)
 

3.1.84.5 Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 1281, normalized size of antiderivative = 7.24 \[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=\text {Too large to display} \]

input

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

output

2/9*((sqrt(2)*(a^2*b^2 + 3*b^4)*cosh(x)^4 + sqrt(2)*(a^2*b^2 + 3*b^4)*sinh 
(x)^4 + 4*sqrt(2)*(a^3*b + 3*a*b^3)*cosh(x)^3 + 4*(sqrt(2)*(a^2*b^2 + 3*b^ 
4)*cosh(x) + sqrt(2)*(a^3*b + 3*a*b^3))*sinh(x)^3 + 2*sqrt(2)*(2*a^4 + 7*a 
^2*b^2 + 3*b^4)*cosh(x)^2 + 2*(3*sqrt(2)*(a^2*b^2 + 3*b^4)*cosh(x)^2 + 6*s 
qrt(2)*(a^3*b + 3*a*b^3)*cosh(x) + sqrt(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4))*si 
nh(x)^2 + 4*sqrt(2)*(a^3*b + 3*a*b^3)*cosh(x) + 4*(sqrt(2)*(a^2*b^2 + 3*b^ 
4)*cosh(x)^3 + 3*sqrt(2)*(a^3*b + 3*a*b^3)*cosh(x)^2 + sqrt(2)*(2*a^4 + 7* 
a^2*b^2 + 3*b^4)*cosh(x) + sqrt(2)*(a^3*b + 3*a*b^3))*sinh(x) + sqrt(2)*(a 
^2*b^2 + 3*b^4))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/2 
7*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b) - 12*(sq 
rt(2)*a*b^3*cosh(x)^4 + sqrt(2)*a*b^3*sinh(x)^4 + 4*sqrt(2)*a^2*b^2*cosh(x 
)^3 + 4*sqrt(2)*a^2*b^2*cosh(x) + sqrt(2)*a*b^3 + 4*(sqrt(2)*a*b^3*cosh(x) 
 + sqrt(2)*a^2*b^2)*sinh(x)^3 + 2*sqrt(2)*(2*a^3*b + a*b^3)*cosh(x)^2 + 2* 
(3*sqrt(2)*a*b^3*cosh(x)^2 + 6*sqrt(2)*a^2*b^2*cosh(x) + sqrt(2)*(2*a^3*b 
+ a*b^3))*sinh(x)^2 + 4*(sqrt(2)*a*b^3*cosh(x)^3 + 3*sqrt(2)*a^2*b^2*cosh( 
x)^2 + sqrt(2)*a^2*b^2 + sqrt(2)*(2*a^3*b + a*b^3)*cosh(x))*sinh(x))*sqrt( 
b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, w 
eierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1 
/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) - 6*(4*a*b^3*cosh(x)^4 + 4*a*b^3* 
sinh(x)^4 + (13*a^2*b^2 - b^4)*cosh(x)^3 + (16*a*b^3*cosh(x) + 13*a^2*b...
 

3.1.84.6 Sympy [F]

\[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \cosh {\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

input

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

output

Integral((a + b*cosh(x))**(-5/2), x)
 

3.1.84.7 Maxima [F]

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

input

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

output

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

3.1.84.8 Giac [F]

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

input

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

output

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

3.1.84.9 Mupad [F(-1)]

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

input

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

output

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