\(\int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx\) [120]

Optimal result

Integrand size = 17, antiderivative size = 231 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=-\frac {2 i \left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}} \] Output:

-2/3*I*(4*A*a*b-B*a^2-3*B*b^2)*(a+b*cosh(x))^(1/2)*EllipticE(I*sinh(1/2*x) 
,2^(1/2)*(b/(a+b))^(1/2))/b/(a^2-b^2)^2/((a+b*cosh(x))/(a+b))^(1/2)+2/3*I* 
(A*b-B*a)*((a+b*cosh(x))/(a+b))^(1/2)*InverseJacobiAM(1/2*I*x,2^(1/2)*(b/( 
a+b))^(1/2))/b/(a^2-b^2)/(a+b*cosh(x))^(1/2)-2/3*(A*b-B*a)*sinh(x)/(a^2-b^ 
2)/(a+b*cosh(x))^(3/2)-2/3*(4*A*a*b-B*a^2-3*B*b^2)*sinh(x)/(a^2-b^2)^2/(a+ 
b*cosh(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.74 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=\frac {2 \left (\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} \left (\left (-4 a A b+a^2 B+3 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (-A b+a B) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{(a-b)^2 b}+\frac {\left (-5 a^2 A b+A b^3+2 a^3 B+2 a b^2 B+b \left (-4 a A b+a^2 B+3 b^2 B\right ) \cosh (x)\right ) \sinh (x)}{\left (a^2-b^2\right )^2}\right )}{3 (a+b \cosh (x))^{3/2}} \] Input:

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

Output:

(2*((I*((a + b*Cosh[x])/(a + b))^(3/2)*((-4*a*A*b + a^2*B + 3*b^2*B)*Ellip 
ticE[(I/2)*x, (2*b)/(a + b)] - (a - b)*(-(A*b) + a*B)*EllipticF[(I/2)*x, ( 
2*b)/(a + b)]))/((a - b)^2*b) + ((-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2*B 
 + b*(-4*a*A*b + a^2*B + 3*b^2*B)*Cosh[x])*Sinh[x])/(a^2 - b^2)^2))/(3*(a 
+ b*Cosh[x])^(3/2))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {-\frac {2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {\frac {2 i \left (a^2-b^2\right ) (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\)

Input:

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

Output:

(-2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) + ((((-2*I) 
*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/ 
(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + ((2*I)*(a^2 - b^2)*(A*b - a* 
B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqr 
t[a + b*Cosh[x]]))/(a^2 - b^2) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sinh[x])/( 
(a^2 - b^2)*Sqrt[a + b*Cosh[x]]))/(3*(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 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. \(796\) vs. \(2(216)=432\).

Time = 11.67 (sec) , antiderivative size = 797, normalized size of antiderivative = 3.45

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 {2 B \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \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 ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \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 ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \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 ) b \right )}{b \sinh \left (\frac {x}{2}\right )^{2} \left (2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b \right ) \sqrt {-\frac {2 b}{a -b}}\, \left (a^{2}-b^{2}\right )}+\frac {2 \left (A b -B a \right ) \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}}}{6 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 {8 \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 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {\left (3 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}}\, \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 b^{2} a -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 {16 a b \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 )}{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 )}{b}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\)	\(797\)
parts	\(\text {Expression too large to display}\)	\(1250\)

Input:

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

Output:

((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)*(-2*B/b/sinh(1/2*x)^2/(2*b*s 
inh(1/2*x)^2+a+b)/(-2*b/(a-b))^(1/2)/(a^2-b^2)*(2*sinh(1/2*x)^4*b+(a+b)*si 
nh(1/2*x)^2)^(1/2)*(2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*b-(-sin 
h(1/2*x)^2)^(1/2)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(co 
sh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*a-(-sinh(1/2*x)^2)^ 
(1/2)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(- 
2*b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*b+2*(-sinh(1/2*x)^2)^(1/2)*(2*b 
/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b)) 
^(1/2),1/2*((-2*a+2*b)/b)^(1/2))*b)+2*(A*b-B*a)/b*(-1/6/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/b 
*(a-b))^2-8/3*sinh(1/2*x)^2*b/(a-b)^2/(a+b)^2*cosh(1/2*x)*a/((2*b*cosh(1/2 
*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3)/(-2* 
b/(a-b))^(1/2)*((2*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)*EllipticF(cosh(1/2*x)*(-2* 
b/(a-b))^(1/2),1/2*((-2*a+2*b)/b)^(1/2))-16/3*a*b/(a+b)^2/(a-b)^2*(-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(co 
sh(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. 2053 vs. \(2 (212) = 424\).

Time = 0.18 (sec) , antiderivative size = 2053, normalized size of antiderivative = 8.89 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

4/9*(sqrt(1/2)*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5 + (2*B*a^3*b 
^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x)^4 + (2*B*a^3*b^2 + A*a^2*b^3 
 - 6*B*a*b^4 + 3*A*b^5)*sinh(x)^4 + 4*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 
 + 3*A*a*b^4)*cosh(x)^3 + 4*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b 
^4 + (2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x))*sinh(x)^3 + 
2*(4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5) 
*cosh(x)^2 + 2*(4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B*a*b 
^4 + 3*A*b^5 + 3*(2*B*a^3*b^2 + A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*cosh(x)^2 
 + 6*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4)*cosh(x))*sinh(x)^2 
+ 4*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4)*cosh(x) + 4*(2*B*a^4 
*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A*a*b^4 + (2*B*a^3*b^2 + A*a^2*b^3 - 6*B* 
a*b^4 + 3*A*b^5)*cosh(x)^3 + 3*(2*B*a^4*b + A*a^3*b^2 - 6*B*a^2*b^3 + 3*A* 
a*b^4)*cosh(x)^2 + (4*B*a^5 + 2*A*a^4*b - 10*B*a^3*b^2 + 7*A*a^2*b^3 - 6*B 
*a*b^4 + 3*A*b^5)*cosh(x))*sinh(x))*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) + 3*sqrt(1/2)*(B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5 + (B*a^2*b^3 - 4 
*A*a*b^4 + 3*B*b^5)*cosh(x)^4 + (B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*sinh(x)^ 
4 + 4*(B*a^3*b^2 - 4*A*a^2*b^3 + 3*B*a*b^4)*cosh(x)^3 + 4*(B*a^3*b^2 - 4*A 
*a^2*b^3 + 3*B*a*b^4 + (B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cosh(x))*sinh(x)^ 
3 + 2*(2*B*a^4*b - 8*A*a^3*b^2 + 7*B*a^2*b^3 - 4*A*a*b^4 + 3*B*b^5)*cos...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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