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

Optimal result

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

[Out]

-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)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 152, 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 = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 \sinh (x) (A b-a B)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i (A b-a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 i B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}} \]

[In]

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

[Out]

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

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 2833

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] + Dist[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]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx=\frac {2 i (a+b) (-A b+a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+2 b (-A b+a B) \sinh (x)}{(a-b) b (a+b) \sqrt {a+b \cosh (x)}} \]

[In]

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

[Out]

((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)]*EllipticF[(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]])

Maple [B] (warning: unable to verify)

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}}\)	\(598\)

[In]

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

[Out]

((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)*Ellipt
icE(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)

Fricas [C] (verification not implemented)

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 )}} \]

[In]

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

[Out]

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)*weierstrassPInv
erse(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) + sqrt(2)*(B*a^2*b - A*a*b^2))*sinh(x) + sqrt(2)*(B*a*b^2 - A*b^3))*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)) + 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))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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