Integrand size = 17, antiderivative size = 233 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {2 i \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{105 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{105 b \sqrt {a+b \cosh (x)}}+\frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x) \] Output:
-2/105*I*(161*A*a^2*b+63*A*b^3+15*B*a^3+145*B*a*b^2)*(a+b*cosh(x))^(1/2)*E llipticE(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))/b/((a+b*cosh(x))/(a+b))^(1 /2)+2/105*I*(a^2-b^2)*(56*A*a*b+15*B*a^2+25*B*b^2)*((a+b*cosh(x))/(a+b))^( 1/2)*InverseJacobiAM(1/2*I*x,2^(1/2)*(b/(a+b))^(1/2))/b/(a+b*cosh(x))^(1/2 )+2/105*(56*A*a*b+15*B*a^2+25*B*b^2)*(a+b*cosh(x))^(1/2)*sinh(x)+2/35*(7*A *b+5*B*a)*(a+b*cosh(x))^(3/2)*sinh(x)+2/7*B*(a+b*cosh(x))^(5/2)*sinh(x)
Time = 0.41 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.87 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {-\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \left (b \left (105 a^3 A+119 a A b^2+135 a^2 b B+25 b^3 B\right ) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \left ((a+b) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )\right )}{b}+(a+b \cosh (x)) \left (154 a A b+90 a^2 B+65 b^2 B+6 b (7 A b+15 a B) \cosh (x)+15 b^2 B \cosh (2 x)\right ) \sinh (x)}{105 \sqrt {a+b \cosh (x)}} \] Input:
Integrate[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*(b*(105*a^3*A + 119*a*A*b^2 + 135*a ^2*b*B + 25*b^3*B)*EllipticF[(I/2)*x, (2*b)/(a + b)] + (161*a^2*A*b + 63*A *b^3 + 15*a^3*B + 145*a*b^2*B)*((a + b)*EllipticE[(I/2)*x, (2*b)/(a + b)] - a*EllipticF[(I/2)*x, (2*b)/(a + b)])))/b + (a + b*Cosh[x])*(154*a*A*b + 90*a^2*B + 65*b^2*B + 6*b*(7*A*b + 15*a*B)*Cosh[x] + 15*b^2*B*Cosh[2*x])*S inh[x])/(105*Sqrt[a + b*Cosh[x]])
Time = 1.43 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.059, Rules used = {3042, 3232, 27, 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.
| \(\displaystyle \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cosh (x))^{3/2} (7 a A+5 b B+(7 A b+5 a B) \cosh (x))dx+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int (a+b \cosh (x))^{3/2} (7 a A+5 b B+(7 A b+5 a B) \cosh (x))dx+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \int \left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^{3/2} \left (7 a A+5 b B+(7 A b+5 a B) \sin \left (i x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cosh (x)} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \cosh (x)\right )dx+\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \cosh (x)} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \cosh (x)\right )dx+\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \sin \left (i x+\frac {\pi }{2}\right )\right )dx\right )\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \cosh (x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{b}\right )+\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}\right )+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 i \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2}+\frac {1}{7} \left (\frac {2}{5} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {2 i \left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\right )\) |
Input:
Int[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(2*B*(a + b*Cosh[x])^(5/2)*Sinh[x])/7 + ((2*(7*A*b + 5*a*B)*(a + b*Cosh[x] )^(3/2)*Sinh[x])/5 + ((((-2*I)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a* 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)*(56*a*A*b + 15*a^2*B + 25*b^2*B )*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt [a + b*Cosh[x]]))/3 + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cosh[ x]]*Sinh[x])/3)/5)/7
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. \(1364\) vs. \(2(214)=428\).
Time = 25.40 (sec) , antiderivative size = 1365, normalized size of antiderivative = 5.86
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1365\) |
| parts | \(\text {Expression too large to display}\) | \(1454\) |
Input:
int((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
2/105*(240*B*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^8*b^3+(168*A*(-2*b /(a-b))^(1/2)*b^3+480*B*(-2*b/(a-b))^(1/2)*a*b^2+360*B*(-2*b/(a-b))^(1/2)* b^3)*sinh(1/2*x)^6*cosh(1/2*x)+(392*A*(-2*b/(a-b))^(1/2)*a*b^2+168*A*(-2*b /(a-b))^(1/2)*b^3+360*B*(-2*b/(a-b))^(1/2)*a^2*b+480*B*(-2*b/(a-b))^(1/2)* a*b^2+280*B*(-2*b/(a-b))^(1/2)*b^3)*sinh(1/2*x)^4*cosh(1/2*x)+(154*A*(-2*b /(a-b))^(1/2)*a^2*b+196*A*(-2*b/(a-b))^(1/2)*a*b^2+42*A*(-2*b/(a-b))^(1/2) *b^3+90*B*(-2*b/(a-b))^(1/2)*a^3+180*B*(-2*b/(a-b))^(1/2)*a^2*b+170*B*(-2* b/(a-b))^(1/2)*a*b^2+80*B*(-2*b/(a-b))^(1/2)*b^3)*sinh(1/2*x)^2*cosh(1/2*x )+105*A*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/b*(a-b))^(1/2))+161* A*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/b*(a-b))^(1/2))+119*A*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)*Ell ipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))+63*A*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/b*(a-b))^(1/2))-322*A*(2*b/(a-b)*si nh(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-126*A*(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))*b^3+15*a^3*B*(2*b/(a-b)*sinh(1...
Leaf count of result is larger than twice the leaf count of optimal. 1123 vs. \(2 (212) = 424\).
Time = 0.10 (sec) , antiderivative size = 1123, normalized size of antiderivative = 4.82 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Too large to display} \] Input:
integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
-1/1260*(16*sqrt(1/2)*((30*B*a^4 + 7*A*a^3*b - 115*B*a^2*b^2 - 231*A*a*b^3 - 75*B*b^4)*cosh(x)^3 + 3*(30*B*a^4 + 7*A*a^3*b - 115*B*a^2*b^2 - 231*A*a *b^3 - 75*B*b^4)*cosh(x)^2*sinh(x) + 3*(30*B*a^4 + 7*A*a^3*b - 115*B*a^2*b ^2 - 231*A*a*b^3 - 75*B*b^4)*cosh(x)*sinh(x)^2 + (30*B*a^4 + 7*A*a^3*b - 1 15*B*a^2*b^2 - 231*A*a*b^3 - 75*B*b^4)*sinh(x)^3)*sqrt(b)*weierstrassPInve rse(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) + 48*sqrt(1/2)*((15*B*a^3*b + 161*A*a^2*b^2 + 145 *B*a*b^3 + 63*A*b^4)*cosh(x)^3 + 3*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b ^3 + 63*A*b^4)*cosh(x)^2*sinh(x) + 3*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a *b^3 + 63*A*b^4)*cosh(x)*sinh(x)^2 + (15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a *b^3 + 63*A*b^4)*sinh(x)^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)) - 3*(15*B*b^4*cosh(x)^6 + 15*B*b^4*sinh(x)^6 + 6*(15*B*a*b^3 + 7*A*b^4)*co sh(x)^5 + 6*(15*B*b^4*cosh(x) + 15*B*a*b^3 + 7*A*b^4)*sinh(x)^5 - 15*B*b^4 + (180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4)*cosh(x)^4 + (225*B*b^4*cosh(x )^2 + 180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4 + 30*(15*B*a*b^3 + 7*A*b^4)* cosh(x))*sinh(x)^4 - 8*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*b^ 4)*cosh(x)^3 + 4*(75*B*b^4*cosh(x)^3 - 30*B*a^3*b - 322*A*a^2*b^2 - 290*B* a*b^3 - 126*A*b^4 + 15*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^2 + (180*B*a^2*b^...
Timed out. \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \] Input:
integrate((a+b*cosh(x))**(5/2)*(A+B*cosh(x)),x)
Output:
Timed out
\[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)
\[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)
Timed out. \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \] Input:
int((A + B*cosh(x))*(a + b*cosh(x))^(5/2),x)
Output:
int((A + B*cosh(x))*(a + b*cosh(x))^(5/2), x)
\[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\left (\int \sqrt {\cosh \left (x \right ) b +a}d x \right ) a^{3}+3 \left (\int \sqrt {\cosh \left (x \right ) b +a}\, \cosh \left (x \right )d x \right ) a^{2} b +\left (\int \sqrt {\cosh \left (x \right ) b +a}\, \cosh \left (x \right )^{3}d x \right ) b^{3}+3 \left (\int \sqrt {\cosh \left (x \right ) b +a}\, \cosh \left (x \right )^{2}d x \right ) a \,b^{2} \] Input:
int((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x)
Output:
int(sqrt(cosh(x)*b + a),x)*a**3 + 3*int(sqrt(cosh(x)*b + a)*cosh(x),x)*a** 2*b + int(sqrt(cosh(x)*b + a)*cosh(x)**3,x)*b**3 + 3*int(sqrt(cosh(x)*b + a)*cosh(x)**2,x)*a*b**2