Integrand size = 17, antiderivative size = 94 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt {a+a \cosh (x)}}+\frac {16}{105} a^2 (7 A+5 B) \sqrt {a+a \cosh (x)} \sinh (x)+\frac {2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+a \cosh (x))^{5/2} \sinh (x) \] Output:
64/105*a^3*(7*A+5*B)*sinh(x)/(a+a*cosh(x))^(1/2)+16/105*a^2*(7*A+5*B)*(a+a *cosh(x))^(1/2)*sinh(x)+2/35*a*(7*A+5*B)*(a+a*cosh(x))^(3/2)*sinh(x)+2/7*B *(a+a*cosh(x))^(5/2)*sinh(x)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.64 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{210} a^2 \sqrt {a (1+\cosh (x))} (1246 A+1040 B+(392 A+505 B) \cosh (x)+6 (7 A+20 B) \cosh (2 x)+15 B \cosh (3 x)) \tanh \left (\frac {x}{2}\right ) \] Input:
Integrate[(a + a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(a^2*Sqrt[a*(1 + Cosh[x])]*(1246*A + 1040*B + (392*A + 505*B)*Cosh[x] + 6* (7*A + 20*B)*Cosh[2*x] + 15*B*Cosh[3*x])*Tanh[x/2])/210
Time = 0.46 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 3230, 3042, 3126, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int (a \cosh (x)+a)^{5/2} (A+B \cosh (x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{5/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {1}{7} (7 A+5 B) \int (\cosh (x) a+a)^{5/2}dx+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}+\frac {1}{7} (7 A+5 B) \int \left (\sin \left (i x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{7} (7 A+5 B) \left (\frac {8}{5} a \int (\cosh (x) a+a)^{3/2}dx+\frac {2}{5} a \sinh (x) (a \cosh (x)+a)^{3/2}\right )+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}+\frac {1}{7} (7 A+5 B) \left (\frac {2}{5} a \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {8}{5} a \int \left (\sin \left (i x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx\right )\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {1}{7} (7 A+5 B) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\cosh (x) a+a}dx+\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}\right )+\frac {2}{5} a \sinh (x) (a \cosh (x)+a)^{3/2}\right )+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}+\frac {1}{7} (7 A+5 B) \left (\frac {2}{5} a \sinh (x) (a \cosh (x)+a)^{3/2}+\frac {8}{5} a \left (\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}+\frac {4}{3} a \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right ) a+a}dx\right )\right )\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {1}{7} (7 A+5 B) \left (\frac {8}{5} a \left (\frac {8 a^2 \sinh (x)}{3 \sqrt {a \cosh (x)+a}}+\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)+a}\right )+\frac {2}{5} a \sinh (x) (a \cosh (x)+a)^{3/2}\right )+\frac {2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2}\) |
Input:
Int[(a + a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]
Output:
(2*B*(a + a*Cosh[x])^(5/2)*Sinh[x])/7 + ((7*A + 5*B)*((2*a*(a + a*Cosh[x]) ^(3/2)*Sinh[x])/5 + (8*a*((8*a^2*Sinh[x])/(3*Sqrt[a + a*Cosh[x]]) + (2*a*S qrt[a + a*Cosh[x]]*Sinh[x])/3))/5))/7
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 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[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 1.58 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {8 \cosh \left (\frac {x}{2}\right ) a^{3} \sinh \left (\frac {x}{2}\right ) \left (30 B \sinh \left (\frac {x}{2}\right )^{6}+\left (21 A +105 B \right ) \sinh \left (\frac {x}{2}\right )^{4}+\left (70 A +140 B \right ) \sinh \left (\frac {x}{2}\right )^{2}+105 A +105 B \right ) \sqrt {2}}{105 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(71\) |
| parts | \(\frac {8 A \,a^{3} \cosh \left (\frac {x}{2}\right ) \sinh \left (\frac {x}{2}\right ) \left (3 \cosh \left (\frac {x}{2}\right )^{4}+4 \cosh \left (\frac {x}{2}\right )^{2}+8\right ) \sqrt {2}}{15 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}+\frac {8 B \cosh \left (\frac {x}{2}\right ) a^{3} \sinh \left (\frac {x}{2}\right ) \left (6 \cosh \left (\frac {x}{2}\right )^{6}+3 \cosh \left (\frac {x}{2}\right )^{4}+4 \cosh \left (\frac {x}{2}\right )^{2}+8\right ) \sqrt {2}}{21 \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\) | \(100\) |
Input:
int((a+cosh(x)*a)^(5/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
Output:
8/105*cosh(1/2*x)*a^3*sinh(1/2*x)*(30*B*sinh(1/2*x)^6+(21*A+105*B)*sinh(1/ 2*x)^4+(70*A+140*B)*sinh(1/2*x)^2+105*A+105*B)*2^(1/2)/(a*cosh(1/2*x)^2)^( 1/2)
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (78) = 156\).
Time = 0.09 (sec) , antiderivative size = 563, normalized size of antiderivative = 5.99 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx =\text {Too large to display} \] Input:
integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")
Output:
1/420*sqrt(1/2)*(15*B*a^2*cosh(x)^7 + 15*B*a^2*sinh(x)^7 + 21*(2*A + 5*B)* a^2*cosh(x)^6 + 35*(10*A + 11*B)*a^2*cosh(x)^5 + 525*(4*A + 3*B)*a^2*cosh( x)^4 + 21*(5*B*a^2*cosh(x) + (2*A + 5*B)*a^2)*sinh(x)^6 - 525*(4*A + 3*B)* a^2*cosh(x)^3 + 7*(45*B*a^2*cosh(x)^2 + 18*(2*A + 5*B)*a^2*cosh(x) + 5*(10 *A + 11*B)*a^2)*sinh(x)^5 - 35*(10*A + 11*B)*a^2*cosh(x)^2 + 35*(15*B*a^2* cosh(x)^3 + 9*(2*A + 5*B)*a^2*cosh(x)^2 + 5*(10*A + 11*B)*a^2*cosh(x) + 15 *(4*A + 3*B)*a^2)*sinh(x)^4 - 21*(2*A + 5*B)*a^2*cosh(x) + 35*(15*B*a^2*co sh(x)^4 + 12*(2*A + 5*B)*a^2*cosh(x)^3 + 10*(10*A + 11*B)*a^2*cosh(x)^2 + 60*(4*A + 3*B)*a^2*cosh(x) - 15*(4*A + 3*B)*a^2)*sinh(x)^3 - 15*B*a^2 + 35 *(9*B*a^2*cosh(x)^5 + 9*(2*A + 5*B)*a^2*cosh(x)^4 + 10*(10*A + 11*B)*a^2*c osh(x)^3 + 90*(4*A + 3*B)*a^2*cosh(x)^2 - 45*(4*A + 3*B)*a^2*cosh(x) - (10 *A + 11*B)*a^2)*sinh(x)^2 + 7*(15*B*a^2*cosh(x)^6 + 18*(2*A + 5*B)*a^2*cos h(x)^5 + 25*(10*A + 11*B)*a^2*cosh(x)^4 + 300*(4*A + 3*B)*a^2*cosh(x)^3 - 225*(4*A + 3*B)*a^2*cosh(x)^2 - 10*(10*A + 11*B)*a^2*cosh(x) - 3*(2*A + 5* B)*a^2)*sinh(x))*sqrt(a/(cosh(x) + sinh(x)))/(cosh(x)^3 + 3*cosh(x)^2*sinh (x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)
Timed out. \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \] Input:
integrate((a+a*cosh(x))**(5/2)*(A+B*cosh(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (78) = 156\).
Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.52 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {1}{60} \, {\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {5}{2} \, x\right )} + 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {3}{2} \, x\right )} + 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (\frac {1}{2} \, x\right )} - 150 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {1}{2} \, x\right )} - 25 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {3}{2} \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-\frac {5}{2} \, x\right )}\right )} A + \frac {1}{168} \, {\left ({\left (3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} + 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {9}{2} \, x\right )} + {\left (7 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-x\right )} + 105 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - 70 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt {2} a^{\frac {5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac {5}{2} \, x\right )}\right )} B \] Input:
integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")
Output:
1/60*(3*sqrt(2)*a^(5/2)*e^(5/2*x) + 25*sqrt(2)*a^(5/2)*e^(3/2*x) + 150*sqr t(2)*a^(5/2)*e^(1/2*x) - 150*sqrt(2)*a^(5/2)*e^(-1/2*x) - 25*sqrt(2)*a^(5/ 2)*e^(-3/2*x) - 3*sqrt(2)*a^(5/2)*e^(-5/2*x))*A + 1/168*((3*sqrt(2)*a^(5/2 )*e^(-x) + 21*sqrt(2)*a^(5/2)*e^(-2*x) + 70*sqrt(2)*a^(5/2)*e^(-3*x) + 210 *sqrt(2)*a^(5/2)*e^(-4*x) - 105*sqrt(2)*a^(5/2)*e^(-5*x) - 7*sqrt(2)*a^(5/ 2)*e^(-6*x))*e^(9/2*x) + (7*sqrt(2)*a^(5/2)*e^(-x) + 105*sqrt(2)*a^(5/2)*e ^(-2*x) - 210*sqrt(2)*a^(5/2)*e^(-3*x) - 70*sqrt(2)*a^(5/2)*e^(-4*x) - 21* sqrt(2)*a^(5/2)*e^(-5*x) - 3*sqrt(2)*a^(5/2)*e^(-6*x))*e^(5/2*x))*B
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.66 \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {1}{840} \, \sqrt {2} a^{4} {\left (\frac {{\left (2100 \, A a^{3} e^{\left (3 \, x\right )} + 1575 \, B a^{3} e^{\left (3 \, x\right )} + 350 \, A a^{3} e^{\left (2 \, x\right )} + 385 \, B a^{3} e^{\left (2 \, x\right )} + 42 \, A a^{3} e^{x} + 105 \, B a^{3} e^{x} + 15 \, B a^{3}\right )} e^{\left (-\frac {7}{2} \, x\right )}}{a^{\frac {9}{2}}} - \frac {15 \, B a^{\frac {67}{2}} e^{\left (\frac {7}{2} \, x\right )} + 42 \, A a^{\frac {67}{2}} e^{\left (\frac {5}{2} \, x\right )} + 105 \, B a^{\frac {67}{2}} e^{\left (\frac {5}{2} \, x\right )} + 350 \, A a^{\frac {67}{2}} e^{\left (\frac {3}{2} \, x\right )} + 385 \, B a^{\frac {67}{2}} e^{\left (\frac {3}{2} \, x\right )} + 2100 \, A a^{\frac {67}{2}} e^{\left (\frac {1}{2} \, x\right )} + 1575 \, B a^{\frac {67}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{35}}\right )} \] Input:
integrate((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")
Output:
-1/840*sqrt(2)*a^4*((2100*A*a^3*e^(3*x) + 1575*B*a^3*e^(3*x) + 350*A*a^3*e ^(2*x) + 385*B*a^3*e^(2*x) + 42*A*a^3*e^x + 105*B*a^3*e^x + 15*B*a^3)*e^(- 7/2*x)/a^(9/2) - (15*B*a^(67/2)*e^(7/2*x) + 42*A*a^(67/2)*e^(5/2*x) + 105* B*a^(67/2)*e^(5/2*x) + 350*A*a^(67/2)*e^(3/2*x) + 385*B*a^(67/2)*e^(3/2*x) + 2100*A*a^(67/2)*e^(1/2*x) + 1575*B*a^(67/2)*e^(1/2*x))/a^35)
Timed out. \[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \] Input:
int((A + B*cosh(x))*(a + a*cosh(x))^(5/2),x)
Output:
int((A + B*cosh(x))*(a + a*cosh(x))^(5/2), x)
\[ \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\cosh \left (x \right )+1}d x \right ) a +2 \left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) a +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )d x \right ) b +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )^{3}d x \right ) b +\left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) a +2 \left (\int \sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )^{2}d x \right ) b \right ) \] Input:
int((a+a*cosh(x))^(5/2)*(A+B*cosh(x)),x)
Output:
sqrt(a)*a**2*(int(sqrt(cosh(x) + 1),x)*a + 2*int(sqrt(cosh(x) + 1)*cosh(x) ,x)*a + int(sqrt(cosh(x) + 1)*cosh(x),x)*b + int(sqrt(cosh(x) + 1)*cosh(x) **3,x)*b + int(sqrt(cosh(x) + 1)*cosh(x)**2,x)*a + 2*int(sqrt(cosh(x) + 1) *cosh(x)**2,x)*b)