Integrand size = 17, antiderivative size = 181 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {2 i \left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) (5 A b+3 a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 b \sqrt {a+b \cosh (x)}}+\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x) \]
2/5*B*(a+b*cosh(x))^(3/2)*sinh(x)+2/15*(5*A*b+3*B*a)*sinh(x)*(a+b*cosh(x)) ^(1/2)-2/15*I*(20*A*a*b+3*B*a^2+9*B*b^2)*(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)+2/15*I*(a^2-b^2)*(5*A*b+3*B*a)*(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*co sh(x))/(a+b))^(1/2)/b/(a+b*cosh(x))^(1/2)
Time = 0.52 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.69 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {2}{15} \sqrt {a+b \cosh (x)} \left (-\frac {i \left (\left (20 a A b+3 a^2 B+9 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (5 A b+3 a B) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+(5 A b+6 a B+3 b B \cosh (x)) \sinh (x)\right ) \]
(2*Sqrt[a + b*Cosh[x]]*(((-I)*((20*a*A*b + 3*a^2*B + 9*b^2*B)*EllipticE[(I /2)*x, (2*b)/(a + b)] - (a - b)*(5*A*b + 3*a*B)*EllipticF[(I/2)*x, (2*b)/( a + b)]))/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + (5*A*b + 6*a*B + 3*b*B*Cosh[ x])*Sinh[x]))/15
Time = 1.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {3042, 3232, 27, 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.
\(\displaystyle \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cosh (x)} (5 a A+3 b B+(5 A b+3 a B) \cosh (x))dx+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \sqrt {a+b \cosh (x)} (5 a A+3 b B+(5 A b+3 a B) \cosh (x))dx+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )} \left (5 a A+3 b B+(5 A b+3 a B) \sin \left (i x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 3232 |
\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \int \sqrt {a+b \cosh (x)}dx}{b}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{b}\right )+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
(2*B*(a + b*Cosh[x])^(3/2)*Sinh[x])/5 + ((((-2*I)*(20*a*A*b + 3*a^2*B + 9* b^2*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)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cos h[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])) /3 + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/3)/5
3.2.8.3.1 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[(-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. \(972\) vs. \(2(197)=394\).
Time = 5.22 (sec) , antiderivative size = 973, normalized size of antiderivative = 5.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(973\) |
parts | \(\text {Expression too large to display}\) | \(1066\) |
2/15*(24*B*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^6*b^2+(20*A*(-2*b/(a -b))^(1/2)*b^2+36*B*(-2*b/(a-b))^(1/2)*a*b+24*B*(-2*b/(a-b))^(1/2)*b^2)*si nh(1/2*x)^4*cosh(1/2*x)+(10*A*(-2*b/(a-b))^(1/2)*a*b+10*A*(-2*b/(a-b))^(1/ 2)*b^2+12*B*(-2*b/(a-b))^(1/2)*a^2+18*B*(-2*b/(a-b))^(1/2)*a*b+6*B*(-2*b/( a-b))^(1/2)*b^2)*sinh(1/2*x)^2*cosh(1/2*x)+15*A*a^2*(2*b/(a-b)*sinh(1/2*x) ^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/( a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+20*b*A*a*(2*b/(a-b)*sinh(1/2*x)^2+(a+b )/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^( 1/2),1/2*(-2*(a-b)/b)^(1/2))+5*b^2*A*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b)) ^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2 *(-2*(a-b)/b)^(1/2))-40*A*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-si nh(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*b+3*a^2*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1 /2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^( 1/2))+12*B*a*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2 )^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+9 *B*b^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)* EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-6*B*(2*b/ (a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(co sh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a^2-18*B*(2*b/(a-b...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 635, normalized size of antiderivative = 3.51 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \sinh \left (x\right )^{2}\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 \, {\left (\sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \sinh \left (x\right )^{2}\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 (3 \, B b^{3} \cosh \left (x\right )^{4} + 3 \, B b^{3} \sinh \left (x\right )^{4} - 3 \, B b^{3} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right ) + 6 \, B a b^{2} + 5 \, A b^{3}\right )} \sinh \left (x\right )^{3} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, B b^{3} \cosh \left (x\right )^{2} - 6 \, B a^{2} b - 40 \, A a b^{2} - 18 \, B b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right )^{3} - 6 \, B a b^{2} - 5 \, A b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{90 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}} \]
-1/90*(4*(sqrt(2)*(6*B*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*cosh(x)^2 + 2*sqrt(2)*(6*B*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*cosh(x)*sinh(x) + sqrt(2)*(6*B*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*sinh(x)^2)*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(2)*(3*B*a^2*b + 20*A* a*b^2 + 9*B*b^3)*cosh(x)^2 + 2*sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)* cosh(x)*sinh(x) + sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)*sinh(x)^2)*sq rt(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*(3*B*b^3*cosh(x)^4 + 3*B*b ^3*sinh(x)^4 - 3*B*b^3 + 2*(6*B*a*b^2 + 5*A*b^3)*cosh(x)^3 + 2*(6*B*b^3*co sh(x) + 6*B*a*b^2 + 5*A*b^3)*sinh(x)^3 - 4*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b ^3)*cosh(x)^2 + 2*(9*B*b^3*cosh(x)^2 - 6*B*a^2*b - 40*A*a*b^2 - 18*B*b^3 + 3*(6*B*a*b^2 + 5*A*b^3)*cosh(x))*sinh(x)^2 - 2*(6*B*a*b^2 + 5*A*b^3)*cosh (x) + 2*(6*B*b^3*cosh(x)^3 - 6*B*a*b^2 - 5*A*b^3 + 3*(6*B*a*b^2 + 5*A*b^3) *cosh(x)^2 - 4*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)*cosh(x))*sinh(x))*sqrt(b *cosh(x) + a))/(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2)
\[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (A + B \cosh {\left (x \right )}\right ) \left (a + b \cosh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2} \,d x \]