\(\int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx\) [119]

Optimal result

Integrand size = 17, antiderivative size = 152 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\frac {2 i (a+b) (-A b+a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (-A b+a B) \sinh (x)}{(a-b) b (a+b) \sqrt {a+b \cosh (x)}} \] Input:

Integrate[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]
 

Output:

((2*I)*(a + b)*(-(A*b) + a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2 
)*x, (2*b)/(a + b)] - (2*I)*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*El 
lipticF[(I/2)*x, (2*b)/(a + b)] + 2*b*(-(A*b) + a*B)*Sinh[x])/((a - b)*b*( 
a + b)*Sqrt[a + b*Cosh[x]])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

Int[(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2),x]
 

Output:

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

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. \(484\) vs. \(2(147)=294\).

Time = 7.42 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.19

method	result	size
default	\(\frac {\sqrt {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (\frac {2 B \sqrt {\frac {2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )}{b \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}-\frac {2 \left (A b -B a \right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b \right )}{b \sinh \left (\frac {x}{2}\right )^{2} \left (2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b \right ) \sqrt {-\frac {2 b}{a -b}}\, \left (a^{2}-b^{2}\right )}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\)	\(485\)
parts	\(-\frac {2 A \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 \left (a -b \right )}{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 \left (a -b \right )}{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 \left (a -b \right )}{b}}}{2}\right ) b \right )}{\sqrt {-\frac {2 b}{a -b}}\, \left (a -b \right ) \left (a +b \right ) \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}+\frac {2 B \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} 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 \left (a -b \right )}{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 \left (a -b \right )}{b}}}{2}\right ) b +2 \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a \right )}{\sqrt {-\frac {2 b}{a -b}}\, \left (a +b \right ) \left (a -b \right ) \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\)	\(598\)

Input:

int((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)*(2*B/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)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/ 
2*((-2*a+2*b)/b)^(1/2))-2*(A*b-B*a)/b/sinh(1/2*x)^2/(2*b*sinh(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)*sinh(1/2*x)^2)^(1/ 
2)*(2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*b-(-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))*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))/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. 601 vs. \(2 (144) = 288\).

Time = 0.09 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.95 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="fricas")
 

Output:

4/3*(sqrt(1/2)*(2*B*a^2*b + A*a*b^2 - 3*B*b^3 + (2*B*a^2*b + A*a*b^2 - 3*B 
*b^3)*cosh(x)^2 + (2*B*a^2*b + A*a*b^2 - 3*B*b^3)*sinh(x)^2 + 2*(2*B*a^3 + 
 A*a^2*b - 3*B*a*b^2)*cosh(x) + 2*(2*B*a^3 + A*a^2*b - 3*B*a*b^2 + (2*B*a^ 
2*b + A*a*b^2 - 3*B*b^3)*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*b^2 - A*b^3 + (B*a*b^2 - A*b^3)*cosh( 
x)^2 + (B*a*b^2 - A*b^3)*sinh(x)^2 + 2*(B*a^2*b - A*a*b^2)*cosh(x) + 2*(B* 
a^2*b - A*a*b^2 + (B*a*b^2 - A*b^3)*cosh(x))*sinh(x))*sqrt(b)*weierstrassZ 
eta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInve 
rse(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*((B*a*b^2 - A*b^3)*cosh(x)^2 + (B*a*b^2 - A* 
b^3)*sinh(x)^2 + (B*a^2*b - A*a*b^2)*cosh(x) + (B*a^2*b - A*a*b^2 + 2*(B*a 
*b^2 - A*b^3)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + a))/(a^2*b^3 - b^5 + (a^2 
*b^3 - b^5)*cosh(x)^2 + (a^2*b^3 - b^5)*sinh(x)^2 + 2*(a^3*b^2 - a*b^4)*co 
sh(x) + 2*(a^3*b^2 - a*b^4 + (a^2*b^3 - b^5)*cosh(x))*sinh(x))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*cosh(x))/(a+b*cosh(x))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="maxima")
 

Output:

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)
 

Giac [F]

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

integrate((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x, algorithm="giac")
 

Output:

integrate((B*cosh(x) + A)/(b*cosh(x) + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((A + B*cosh(x))/(a + b*cosh(x))^(3/2),x)
 

Output:

int((A + B*cosh(x))/(a + b*cosh(x))^(3/2), x)
 

Reduce [F]

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

int((A+B*cosh(x))/(a+b*cosh(x))^(3/2),x)
 

Output:

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