Integrand size = 17, antiderivative size = 138 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=-\frac {2 i (3 A b+a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}+\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x) \] Output:
-2/3*I*(3*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+b*cosh(x))/(a+b))^(1/2)+2/3*I*(a^2-b^2)*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*co sh(x))^(1/2)+2/3*B*(a+b*cosh(x))^(1/2)*sinh(x)
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\frac {-2 i (a+b) (3 A b+a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b B (a+b \cosh (x)) \sinh (x)}{3 b \sqrt {a+b \cosh (x)}} \] Input:
Integrate[Sqrt[a + b*Cosh[x]]*(A + B*Cosh[x]),x]
Output:
((-2*I)*(a + b)*(3*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*B*(a + b*Cosh[x])*Sinh[x])/(3*b*Sqrt [a + b*Cosh[x]])
Time = 0.77 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3042, 3232, 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.
| \(\displaystyle \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin \left (\frac {\pi }{2}+i x\right )} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2}{3} \int \frac {3 a A+b B+(3 A b+a B) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {3 a A+b B+(3 A b+a B) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {3 a A+b B+(3 A b+a B) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{3} \left (\frac {(a B+3 A b) \int \sqrt {a+b \cosh (x)}dx}{b}-\frac {B \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{b}\right )+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}-\frac {B \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {B \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {(a B+3 A b) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {B \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {B \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\) |
Input:
Int[Sqrt[a + b*Cosh[x]]*(A + B*Cosh[x]),x]
Output:
(((-2*I)*(3*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*C osh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]] ))/3 + (2*B*Sqrt[a + b*Cosh[x]]*Sinh[x])/3
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[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*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] && GtQ[m, 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(127)=254\).
Time = 14.03 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.34
| method | result | size |
| parts | \(\frac {2 A \left (a \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 \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 )-2 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 )\right ) \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b}{a -b}}\, \sqrt {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\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}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}+\frac {2 B \left (4 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{4} b +2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} a +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 {\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 {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{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}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\) | \(599\) |
| default | \(\frac {2 \left (4 B \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{4} b +2 B \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} a +2 B \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b +3 A a \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 {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 )+3 A b \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 {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 )-6 A \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 ) b +B a \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 {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 b \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 {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 )-2 B \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 {\left (2 b \cosh \left (\frac {x}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{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}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\) | \(613\) |
Input:
int((a+b*cosh(x))^(1/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
2*A*(a*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))+b* EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))-2*b*Ellip ticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2)))*(-sinh(1/2*x) ^2)^(1/2)*((2*b*cosh(1/2*x)^2+a-b)/(a-b))^(1/2)*((2*b*cosh(1/2*x)^2+a-b)*s inh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x )^2)^(1/2)/sinh(1/2*x)/(2*b*sinh(1/2*x)^2+a+b)^(1/2)+2/3*B*(4*cosh(1/2*x)* (-2*b/(a-b))^(1/2)*sinh(1/2*x)^4*b+2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1 /2*x)^2*a+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/b*(a-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/b*(a-b))^(1/2))*b-2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/ 2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2 /b*(a-b))^(1/2))*a)*((2*b*cosh(1/2*x)^2+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a -b))^(1/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/sinh(1/2*x)/(2*b* sinh(1/2*x)^2+a+b)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (126) = 252\).
Time = 0.08 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.31 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=-\frac {4 \, \sqrt {\frac {1}{2}} {\left ({\left (2 \, B a^{2} - 3 \, A a b - 3 \, B b^{2}\right )} \cosh \left (x\right ) + {\left (2 \, B a^{2} - 3 \, A a b - 3 \, B b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, \sqrt {\frac {1}{2}} {\left ({\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + {\left (B a b + 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (B b^{2} \cosh \left (x\right )^{2} + B b^{2} \sinh \left (x\right )^{2} - B b^{2} - 2 \, {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (B b^{2} \cosh \left (x\right ) - B a b - 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{9 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \] Input:
integrate((a+b*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
-1/9*(4*sqrt(1/2)*((2*B*a^2 - 3*A*a*b - 3*B*b^2)*cosh(x) + (2*B*a^2 - 3*A* a*b - 3*B*b^2)*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) + 12*sqrt(1/2)*((B*a*b + 3*A*b^2)*cosh(x) + (B*a*b + 3*A*b^2)*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)) - 3*(B*b^2*cosh(x)^2 + B*b^2*sin h(x)^2 - B*b^2 - 2*(B*a*b + 3*A*b^2)*cosh(x) + 2*(B*b^2*cosh(x) - B*a*b - 3*A*b^2)*sinh(x))*sqrt(b*cosh(x) + a))/(b^2*cosh(x) + b^2*sinh(x))
\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int \left (A + B \cosh {\left (x \right )}\right ) \sqrt {a + b \cosh {\left (x \right )}}\, dx \] Input:
integrate((a+b*cosh(x))**(1/2)*(A+B*cosh(x)),x)
Output:
Integral((A + B*cosh(x))*sqrt(a + b*cosh(x)), x)
\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} \sqrt {b \cosh \left (x\right ) + a} \,d x } \] Input:
integrate((a+b*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)
\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} \sqrt {b \cosh \left (x\right ) + a} \,d x } \] Input:
integrate((a+b*cosh(x))^(1/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)
Timed out. \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,\sqrt {a+b\,\mathrm {cosh}\left (x\right )} \,d x \] Input:
int((A + B*cosh(x))*(a + b*cosh(x))^(1/2),x)
Output:
int((A + B*cosh(x))*(a + b*cosh(x))^(1/2), x)
\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\left (\int \sqrt {\cosh \left (x \right ) b +a}d x \right ) a +\left (\int \sqrt {\cosh \left (x \right ) b +a}\, \cosh \left (x \right )d x \right ) b \] Input:
int((a+b*cosh(x))^(1/2)*(A+B*cosh(x)),x)
Output:
int(sqrt(cosh(x)*b + a),x)*a + int(sqrt(cosh(x)*b + a)*cosh(x),x)*b