Integrand size = 17, antiderivative size = 152 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \]
-2*(A*b-B*a)*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^(1/2)-2*I*(A*b-B*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)/b/(a^2-b^2)/((a+b*cosh(x))/(a+b))^(1/2)-2*I*B*(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)/b/(a+b*cosh(x))^(1/2)
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\frac {2 i (a+b) (-A b+a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (-A b+a B) \sinh (x)}{(a-b) b (a+b) \sqrt {a+b \cosh (x)}} \]
((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]])
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 |
\(\displaystyle -\frac {2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}+\frac {-\frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}}{a^2-b^2}\) |
(((-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]])
3.2.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(484\) vs. \(2(178)=356\).
Time = 2.76 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.19
method | result | size |
default | \(\frac {\sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (\frac {2 B \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )}{b \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}-\frac {2 \left (b A -B a \right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right ) b \right )}{b \sinh \left (\frac {x}{2}\right )^{2} \left (2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b \right ) \sqrt {-\frac {2 b}{a -b}}\, \left (a^{2}-b^{2}\right )}\right )}{\sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\) | \(485\) |
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 \sinh \left (\frac {x}{2}\right )^{2} b +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 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\) | \(598\) |
((2*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)*(2*B/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)*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*sinh(1/2*x)^2*b+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*sinh(1/2*x)^2*b+a+b)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 639, normalized size of antiderivative = 4.20 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} - 3 \, B b^{3}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} - 3 \, B b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (2 \, B a^{3} + A a^{2} b - 3 \, B a b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (\sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} - 3 \, B b^{3}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{3} + A a^{2} b - 3 \, B a b^{2}\right )}\right )} \sinh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{2} b + A a b^{2} - 3 \, B b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 3 \, {\left (\sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (B a^{2} b - A a b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (\sqrt {2} {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (B a^{2} b - A a b^{2}\right )}\right )} \sinh \left (x\right ) + \sqrt {2} {\left (B a b^{2} - A b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )^{2} + {\left (B a b^{2} - A b^{3}\right )} \sinh \left (x\right )^{2} + {\left (B a^{2} b - A a b^{2}\right )} \cosh \left (x\right ) + {\left (B a^{2} b - A a b^{2} + 2 \, {\left (B a b^{2} - A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
2/3*((sqrt(2)*(2*B*a^2*b + A*a*b^2 - 3*B*b^3)*cosh(x)^2 + sqrt(2)*(2*B*a^2 *b + A*a*b^2 - 3*B*b^3)*sinh(x)^2 + 2*sqrt(2)*(2*B*a^3 + A*a^2*b - 3*B*a*b ^2)*cosh(x) + 2*(sqrt(2)*(2*B*a^2*b + A*a*b^2 - 3*B*b^3)*cosh(x) + sqrt(2) *(2*B*a^3 + A*a^2*b - 3*B*a*b^2))*sinh(x) + sqrt(2)*(2*B*a^2*b + A*a*b^2 - 3*B*b^3))*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(2)*( B*a*b^2 - A*b^3)*cosh(x)^2 + sqrt(2)*(B*a*b^2 - A*b^3)*sinh(x)^2 + 2*sqrt( 2)*(B*a^2*b - A*a*b^2)*cosh(x) + 2*(sqrt(2)*(B*a*b^2 - A*b^3)*cosh(x) + sq rt(2)*(B*a^2*b - A*a*b^2))*sinh(x) + sqrt(2)*(B*a*b^2 - A*b^3))*sqrt(b)*we ierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weiers trassPInverse(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*((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)*cosh(x) + 2*(a^3*b^2 - a*b^4 + (a^2*b^3 - b^5)*cosh(x))*sinh(x))
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\int { \frac {B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\int { \frac {B \cosh \left (x\right ) + A}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]