Integrand size = 10, antiderivative size = 153 \[ \int (a+b \cosh (x))^{5/2} \, dx=-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 \sqrt {a+b \cosh (x)}}+\frac {16}{15} a b \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} b (a+b \cosh (x))^{3/2} \sinh (x) \]
2/5*b*(a+b*cosh(x))^(3/2)*sinh(x)+16/15*a*b*sinh(x)*(a+b*cosh(x))^(1/2)-2/ 15*I*(23*a^2+9*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)/((a+b*cosh(x))/(a+b))^(1/ 2)+16/15*I*a*(a^2-b^2)*(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)/(a+b*cosh(x))^ (1/2)
Time = 0.41 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.98 \[ \int (a+b \cosh (x))^{5/2} \, dx=\frac {-2 i \left (23 a^3+23 a^2 b+9 a b^2+9 b^3\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+b \left (22 a^2+3 b^2+28 a b \cosh (x)+3 b^2 \cosh (2 x)\right ) \sinh (x)}{15 \sqrt {a+b \cosh (x)}} \]
((-2*I)*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[(a + b*Cosh[x])/(a + b) ]*EllipticE[(I/2)*x, (2*b)/(a + b)] + (16*I)*a*(a^2 - b^2)*Sqrt[(a + b*Cos h[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)] + b*(22*a^2 + 3*b^2 + 28* a*b*Cosh[x] + 3*b^2*Cosh[2*x])*Sinh[x])/(15*Sqrt[a + b*Cosh[x]])
Time = 0.90 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 3135, 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.
| \(\displaystyle \int (a+b \cosh (x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cosh (x)} \left (5 a^2+8 b \cosh (x) a+3 b^2\right )dx+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \sqrt {a+b \cosh (x)} \left (5 a^2+8 b \cosh (x) a+3 b^2\right )dx+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )} \left (5 a^2+8 b \sin \left (i x+\frac {\pi }{2}\right ) a+3 b^2\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx+\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx+\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cosh (x)}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx\right )+\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{5} b \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \sinh (x) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {16 i a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{\sqrt {a+b \cosh (x)}}-\frac {2 i \left (23 a^2+9 b^2\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\) |
(2*b*(a + b*Cosh[x])^(3/2)*Sinh[x])/5 + ((((-2*I)*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/Sqrt[(a + b*Cosh[x])/(a + b)] + ((16*I)*a*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)* x, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]])/3 + (16*a*b*Sqrt[a + b*Cosh[x]]*Si nh[x])/3)/5
3.1.79.3.1 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[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[1/n Int[(a + b* Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x] , x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
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. \(684\) vs. \(2(169)=338\).
Time = 6.49 (sec) , antiderivative size = 685, normalized size of antiderivative = 4.48
| method | result | size |
| default | \(\frac {2 \left (24 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{6} b^{3}+\left (56 \sqrt {-\frac {2 b}{a -b}}\, a \,b^{2}+24 \sqrt {-\frac {2 b}{a -b}}\, b^{3}\right ) \sinh \left (\frac {x}{2}\right )^{4} \cosh \left (\frac {x}{2}\right )+\left (22 \sqrt {-\frac {2 b}{a -b}}\, a^{2} b +28 \sqrt {-\frac {2 b}{a -b}}\, a \,b^{2}+6 \sqrt {-\frac {2 b}{a -b}}\, b^{3}\right ) \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )+15 a^{3} \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 )+23 a^{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 {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 )+17 a \,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 {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 )+9 b^{3} \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 )-46 \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^{2} b -18 \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^{3}\right ) \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{15 \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}}\) | \(685\) |
2/15*(24*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^6*b^3+(56*(-2*b/(a-b)) ^(1/2)*a*b^2+24*(-2*b/(a-b))^(1/2)*b^3)*sinh(1/2*x)^4*cosh(1/2*x)+(22*(-2* b/(a-b))^(1/2)*a^2*b+28*(-2*b/(a-b))^(1/2)*a*b^2+6*(-2*b/(a-b))^(1/2)*b^3) *sinh(1/2*x)^2*cosh(1/2*x)+15*a^3*(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))+23*a^2*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-s inh(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))+17*a*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))+9*b^3*(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))-46*(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*b-18*(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))*b^3)*((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)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.03 \[ \int (a+b \cosh (x))^{5/2} \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (a^{3} - 33 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} - 33 \, a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} - 33 \, a b^{2}\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 (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b + 9 \, 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^{3} \cosh \left (x\right )^{4} + 3 \, b^{3} \sinh \left (x\right )^{4} + 22 \, a b^{2} \cosh \left (x\right )^{3} - 22 \, a b^{2} \cosh \left (x\right ) + 2 \, {\left (6 \, b^{3} \cosh \left (x\right ) + 11 \, a b^{2}\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 33 \, a b^{2} \cosh \left (x\right ) - 46 \, a^{2} b - 18 \, b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, b^{3} \cosh \left (x\right )^{3} + 33 \, a b^{2} \cosh \left (x\right )^{2} - 11 \, a b^{2} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{90 \, {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2}\right )}} \]
-1/90*(4*(sqrt(2)*(a^3 - 33*a*b^2)*cosh(x)^2 + 2*sqrt(2)*(a^3 - 33*a*b^2)* cosh(x)*sinh(x) + sqrt(2)*(a^3 - 33*a*b^2)*sinh(x)^2)*sqrt(b)*weierstrassP Inverse(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*cos h(x) + 3*b*sinh(x) + 2*a)/b) + 12*(sqrt(2)*(23*a^2*b + 9*b^3)*cosh(x)^2 + 2*sqrt(2)*(23*a^2*b + 9*b^3)*cosh(x)*sinh(x) + sqrt(2)*(23*a^2*b + 9*b^3)* sinh(x)^2)*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*(3*b^3*cosh(x )^4 + 3*b^3*sinh(x)^4 + 22*a*b^2*cosh(x)^3 - 22*a*b^2*cosh(x) + 2*(6*b^3*c osh(x) + 11*a*b^2)*sinh(x)^3 - 3*b^3 - 4*(23*a^2*b + 9*b^3)*cosh(x)^2 + 2* (9*b^3*cosh(x)^2 + 33*a*b^2*cosh(x) - 46*a^2*b - 18*b^3)*sinh(x)^2 + 2*(6* b^3*cosh(x)^3 + 33*a*b^2*cosh(x)^2 - 11*a*b^2 - 4*(23*a^2*b + 9*b^3)*cosh( x))*sinh(x))*sqrt(b*cosh(x) + a))/(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*s inh(x)^2)
\[ \int (a+b \cosh (x))^{5/2} \, dx=\int \left (a + b \cosh {\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int (a+b \cosh (x))^{5/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (a+b \cosh (x))^{5/2} \, dx=\int { {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+b \cosh (x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \]