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

Optimal result

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

[Out]

-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*sinh(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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2743, 2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]

[In]

Int[(a + b*Cosh[x])^(-5/2),x]

[Out]

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

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2743

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] + Dist[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] && Integ
erQ[2*n]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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 2833

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3 a}{2}+\frac {1}{2} 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}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2+b^2\right )+a b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )^2} \\ & = -\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)}}+\frac {(4 a) \int \sqrt {a+b \cosh (x)} \, dx}{3 \left (a^2-b^2\right )^2}-\frac {\int \frac {1}{\sqrt {a+b \cosh (x)}} \, 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}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {\left (4 a \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 \left (a^2-b^2\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 )}{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)}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(a + b*Cosh[x])^(-5/2),x]

[Out]

((-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)*EllipticF[(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))

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

[In]

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

[Out]

((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)*(Ellipti
cF(cosh(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)

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 \[ \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*sqrt(2)*(a^3*b + 3*a*b^3)*cosh(x) +
sqrt(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4))*sinh(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
, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b) - 12*(sqrt(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, weierstrassPInverse(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^2 - b^4)*sinh(x)^3 + 4*(2*a^3*b + a*b^3)*cosh(x)^2 + (24*a*b^3
*cosh(x)^2 + 8*a^3*b + 4*a*b^3 + 3*(13*a^2*b^2 - b^4)*cosh(x))*sinh(x)^2 + (3*a^2*b^2 + b^4)*cosh(x) + (16*a*b
^3*cosh(x)^3 + 3*a^2*b^2 + b^4 + 3*(13*a^2*b^2 - b^4)*cosh(x)^2 + 8*(2*a^3*b + a*b^3)*cosh(x))*sinh(x))*sqrt(b
*cosh(x) + a))/(a^4*b^3 - 2*a^2*b^5 + b^7 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x)^4 + (a^4*b^3 - 2*a^2*b^5 + b^7
)*sinh(x)^4 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cosh(x)^3 + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^
5 + b^7)*cosh(x))*sinh(x)^3 + 2*(2*a^6*b - 3*a^4*b^3 + b^7)*cosh(x)^2 + 2*(2*a^6*b - 3*a^4*b^3 + b^7 + 3*(a^4*
b^3 - 2*a^2*b^5 + b^7)*cosh(x)^2 + 6*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^2 + 4*(a^5*b^2 - 2*a^3*b^4
+ a*b^6)*cosh(x) + 4*(a^5*b^2 - 2*a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^5 + b^7)*cosh(x)^3 + 3*(a^5*b^2 - 2*a^
3*b^4 + a*b^6)*cosh(x)^2 + (2*a^6*b - 3*a^4*b^3 + b^7)*cosh(x))*sinh(x))

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*cosh(x))**(-5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*cosh(x) + a)^(-5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((b*cosh(x) + a)^(-5/2), x)

Mupad [F(-1)]

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

[In]

int(1/(a + b*cosh(x))^(5/2),x)

[Out]

int(1/(a + b*cosh(x))^(5/2), x)