\(\int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx\) [109]

Optimal result

Integrand size = 17, antiderivative size = 138 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=-\frac {2 i (3 A b+a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}+\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x) \]

[Out]

2/3*B*sinh(x)*(a+b*cosh(x))^(1/2)-2/3*I*(3*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+b*cosh(x))/(a+b))^(1/2)+2/3*I*(a^2-b^2)*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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)} \]

[In]

Int[Sqrt[a + b*Cosh[x]]*(A + B*Cosh[x]),x]

[Out]

(((-2*I)/3)*(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)/3)*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b
*Cosh[x]]) + (2*B*Sqrt[a + b*Cosh[x]]*Sinh[x])/3

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{3} \int \frac {\frac {1}{2} (3 a A+b B)+\frac {1}{2} (3 A b+a B) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)-\frac {\left (\left (a^2-b^2\right ) B\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{3 b}+\frac {(3 A b+a B) \int \sqrt {a+b \cosh (x)} \, dx}{3 b} \\ & = \frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)+\frac {\left ((3 A b+a B) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 b \sqrt {a+b \cosh (x)}} \\ & = -\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) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\frac {-2 i (a+b) (3 A b+a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b B (a+b \cosh (x)) \sinh (x)}{3 b \sqrt {a+b \cosh (x)}} \]

[In]

Integrate[Sqrt[a + b*Cosh[x]]*(A + B*Cosh[x]),x]

[Out]

((-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)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + 2*b*B*(a + b*Cosh[x])*Sinh[x])/(3*b*Sq
rt[a + b*Cosh[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(158)=316\).

Time = 5.14 (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 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +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 \sinh \left (\frac {x}{2}\right )^{2} b +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 \cosh \left (\frac {x}{2}\right )^{2} b +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 \sinh \left (\frac {x}{2}\right )^{2} b +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 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 )-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 \cosh \left (\frac {x}{2}\right )^{2} b +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 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\)	\(613\)

[In]

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

[Out]

2*A*(a*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+b*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(
1/2),1/2*(-2*(a-b)/b)^(1/2))-2*b*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2)))*(-sinh(1/2*
x)^2)^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+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*sinh(1/2*x)^2*b+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*(a-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-b)/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*(a-b)/b)^(1/2))*a)*((2*cosh
(1/2*x)^2*b+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*sinh(1/2*x)^2*b+a+b)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.36 \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b - 3 \, B b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (2 \, B a^{2} - 3 \, A a b - 3 \, B b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 6 \, {\left (\sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (B a b + 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (B b^{2} \cosh \left (x\right )^{2} + B b^{2} \sinh \left (x\right )^{2} - B b^{2} - 2 \, {\left (B a b + 3 \, A b^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (B b^{2} \cosh \left (x\right ) - B a b - 3 \, A b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{9 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

-1/9*(2*(sqrt(2)*(2*B*a^2 - 3*A*a*b - 3*B*b^2)*cosh(x) + sqrt(2)*(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) + 6*(sqrt(2)*(B*a*b + 3*A*b^2)*cosh(x) + sqrt(2)*(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*sinh(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))

Sympy [F]

\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int \left (A + B \cosh {\left (x \right )}\right ) \sqrt {a + b \cosh {\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral((A + B*cosh(x))*sqrt(a + b*cosh(x)), x)

Maxima [F]

\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} \sqrt {b \cosh \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)

Giac [F]

\[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} \sqrt {b \cosh \left (x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate((B*cosh(x) + A)*sqrt(b*cosh(x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,\sqrt {a+b\,\mathrm {cosh}\left (x\right )} \,d x \]

[In]

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

[Out]

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