\(\int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx\) [85]

Optimal result

Integrand size = 10, antiderivative size = 227 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {16 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}-\frac {16 a b \sinh (x)}{15 \left (a^2-b^2\right )^2 (a+b \cosh (x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{15 \left (a^2-b^2\right )^3 \sqrt {a+b \cosh (x)}} \] Output:

-2/15*I*(23*a^2+9*b^2)*(a+b*cosh(x))^(1/2)*EllipticE(I*sinh(1/2*x),2^(1/2) 
*(b/(a+b))^(1/2))/(a^2-b^2)^3/((a+b*cosh(x))/(a+b))^(1/2)+16/15*I*a*((a+b* 
cosh(x))/(a+b))^(1/2)*InverseJacobiAM(1/2*I*x,2^(1/2)*(b/(a+b))^(1/2))/(a^ 
2-b^2)^2/(a+b*cosh(x))^(1/2)-2/5*b*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(5/2)-1 
6/15*a*b*sinh(x)/(a^2-b^2)^2/(a+b*cosh(x))^(3/2)-2/15*b*(23*a^2+9*b^2)*sin 
h(x)/(a^2-b^2)^3/(a+b*cosh(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\frac {2 \left (-\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+8 a (-a+b) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (34 a^4-5 a^2 b^2+3 b^4+2 a b \left (27 a^2+5 b^2\right ) \cosh (x)+b^2 \left (23 a^2+9 b^2\right ) \cosh ^2(x)\right ) \sinh (x)}{\left (-a^2+b^2\right )^3}\right )}{15 (a+b \cosh (x))^{5/2}} \] Input:

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

Output:

(2*(((-I)*((a + b*Cosh[x])/(a + b))^(5/2)*((23*a^2 + 9*b^2)*EllipticE[(I/2 
)*x, (2*b)/(a + b)] + 8*a*(-a + b)*EllipticF[(I/2)*x, (2*b)/(a + b)]))/(a 
- b)^3 + (b*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*Cosh[x] + 
 b^2*(23*a^2 + 9*b^2)*Cosh[x]^2)*Sinh[x])/(-a^2 + b^2)^3))/(15*(a + b*Cosh 
[x])^(5/2))
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {3042, 3143, 27, 3042, 3233, 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.

\(\displaystyle \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{7/2}}dx\)

\(\Big \downarrow \) 3143

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {2 \int -\frac {3 \left (5 a^2+3 b^2\right )-8 a b \cosh (x)}{2 (a+b \cosh (x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \cosh (x)}{(a+b \cosh (x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \sin \left (i x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cosh (x)}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 b \sinh (x)}{5 \left (a^2-b^2\right ) (a+b \cosh (x))^{5/2}}+\frac {-\frac {16 a b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 b \left (23 a^2+9 b^2\right ) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\)

Input:

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

Output:

(-2*b*Sinh[x])/(5*(a^2 - b^2)*(a + b*Cosh[x])^(5/2)) + ((-16*a*b*Sinh[x])/ 
(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) + ((((-2*I)*(23*a^2 + 9*b^2)*Sqrt[a 
+ b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + 
b)] + ((16*I)*a*(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) - (2*b*(23*a^2 + 9*b^2 
)*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*Cosh[x]]))/(3*(a^2 - b^2)))/(5*(a^2 - b 
^2))
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(208)=416\).

Time = 2.71 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.49

method	result	size
default	\(\frac {\sqrt {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+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}}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{3}}-\frac {8 a \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}}}{15 b \left (a +b \right )^{2} \left (a -b \right )^{2} \left (\cosh \left (\frac {x}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{2}}-\frac {4 \sinh \left (\frac {x}{2}\right )^{2} b \cosh \left (\frac {x}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {2 b \cosh \left (\frac {x}{2}\right )^{2}+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 (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\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 {8 b \left (23 a^{2}+9 b^{2}\right ) \left (-a +b \right ) \sqrt {\frac {2 b \cosh \left (\frac {x}{2}\right )^{2}+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 )}{15 \left (a +b \right )^{3} \left (a -b \right )^{3} \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 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\)	\(566\)

Input:

int(1/(a+b*cosh(x))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)*(-1/10/b^2/(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/b* 
(a-b))^3-8/15*a/b/(a+b)^2/(a-b)^2*cosh(1/2*x)*(2*sinh(1/2*x)^4*b+(a+b)*sin 
h(1/2*x)^2)^(1/2)/(cosh(1/2*x)^2+1/2/b*(a-b))^2-4/15*sinh(1/2*x)^2*b/(a-b) 
^3/(a+b)^3*cosh(1/2*x)*(23*a^2+9*b^2)/((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x) 
^2)^(1/2)+2*(15*a^2-8*a*b+9*b^2)/(15*a^5+15*a^4*b-30*a^3*b^2-30*a^2*b^3+15 
*a*b^4+15*b^5)/(-2*b/(a-b))^(1/2)*((2*b*cosh(1/2*x)^2+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)*Ellipti 
cF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-8/15*b*(23*a^2 
+9*b^2)/(a+b)^3/(a-b)^3*(-a+b)/(-2*b/(a-b))^(1/2)*((2*b*cosh(1/2*x)^2+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(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*b*sinh(1/2*x)^2+a+b)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3129 vs. \(2 (204) = 408\).

Time = 0.15 (sec) , antiderivative size = 3129, normalized size of antiderivative = 13.78 \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cosh(x))**(7/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*cosh(x))^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{7/2}} \,d x \] Input:

int(1/(a + b*cosh(x))^(7/2),x)
 

Output:

int(1/(a + b*cosh(x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \cosh (x))^{7/2}} \, dx=\int \frac {\sqrt {\cosh \left (x \right ) b +a}}{\cosh \left (x \right )^{4} b^{4}+4 \cosh \left (x \right )^{3} a \,b^{3}+6 \cosh \left (x \right )^{2} a^{2} b^{2}+4 \cosh \left (x \right ) a^{3} b +a^{4}}d x \] Input:

int(1/(a+b*cosh(x))^(7/2),x)
 

Output:

int(sqrt(cosh(x)*b + a)/(cosh(x)**4*b**4 + 4*cosh(x)**3*a*b**3 + 6*cosh(x) 
**2*a**2*b**2 + 4*cosh(x)*a**3*b + a**4),x)