Integrand size = 12, antiderivative size = 183 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {1}{8} a \left (8 a^4+40 a^2 b^2+15 b^4\right ) x+\frac {b \left (107 a^4+192 a^2 b^2+16 b^4\right ) \sinh (c+d x)}{30 d}+\frac {7 a b^2 \left (22 a^2+23 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{120 d}+\frac {b \left (47 a^2+16 b^2\right ) (a+b \cosh (c+d x))^2 \sinh (c+d x)}{60 d}+\frac {9 a b (a+b \cosh (c+d x))^3 \sinh (c+d x)}{20 d}+\frac {b (a+b \cosh (c+d x))^4 \sinh (c+d x)}{5 d} \] Output:
1/8*a*(8*a^4+40*a^2*b^2+15*b^4)*x+1/30*b*(107*a^4+192*a^2*b^2+16*b^4)*sinh (d*x+c)/d+7/120*a*b^2*(22*a^2+23*b^2)*cosh(d*x+c)*sinh(d*x+c)/d+1/60*b*(47 *a^2+16*b^2)*(a+b*cosh(d*x+c))^2*sinh(d*x+c)/d+9/20*a*b*(a+b*cosh(d*x+c))^ 3*sinh(d*x+c)/d+1/5*b*(a+b*cosh(d*x+c))^4*sinh(d*x+c)/d
Time = 0.71 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {60 a \left (8 a^4+40 a^2 b^2+15 b^4\right ) (c+d x)+300 b \left (8 a^4+12 a^2 b^2+b^4\right ) \sinh (c+d x)+600 a b^2 \left (2 a^2+b^2\right ) \sinh (2 (c+d x))+50 b^3 \left (8 a^2+b^2\right ) \sinh (3 (c+d x))+75 a b^4 \sinh (4 (c+d x))+6 b^5 \sinh (5 (c+d x))}{480 d} \] Input:
Integrate[(a + b*Cosh[c + d*x])^5,x]
Output:
(60*a*(8*a^4 + 40*a^2*b^2 + 15*b^4)*(c + d*x) + 300*b*(8*a^4 + 12*a^2*b^2 + b^4)*Sinh[c + d*x] + 600*a*b^2*(2*a^2 + b^2)*Sinh[2*(c + d*x)] + 50*b^3* (8*a^2 + b^2)*Sinh[3*(c + d*x)] + 75*a*b^4*Sinh[4*(c + d*x)] + 6*b^5*Sinh[ 5*(c + d*x)])/(480*d)
Time = 0.81 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3135, 3042, 3232, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int (a+b \cosh (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^5dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {1}{5} \int (a+b \cosh (c+d x))^3 \left (5 a^2+9 b \cosh (c+d x) a+4 b^2\right )dx+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac {1}{5} \int \left (a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^3 \left (5 a^2+9 b \sin \left (i c+i d x+\frac {\pi }{2}\right ) a+4 b^2\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int (a+b \cosh (c+d x))^2 \left (a \left (20 a^2+43 b^2\right )+b \left (47 a^2+16 b^2\right ) \cosh (c+d x)\right )dx+\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}\right )+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac {1}{5} \left (\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}+\frac {1}{4} \int \left (a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^2 \left (a \left (20 a^2+43 b^2\right )+b \left (47 a^2+16 b^2\right ) \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )dx\right )\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \cosh (c+d x)) \left (60 a^4+223 b^2 a^2+7 b \left (22 a^2+23 b^2\right ) \cosh (c+d x) a+32 b^4\right )dx+\frac {b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{3 d}\right )+\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}\right )+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}+\frac {1}{5} \left (\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}+\frac {1}{4} \left (\frac {b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{3 d}+\frac {1}{3} \int \left (a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )\right ) \left (60 a^4+223 b^2 a^2+7 b \left (22 a^2+23 b^2\right ) \sin \left (i c+i d x+\frac {\pi }{2}\right ) a+32 b^4\right )dx\right )\right )\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {b \left (47 a^2+16 b^2\right ) \sinh (c+d x) (a+b \cosh (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {7 a b^2 \left (22 a^2+23 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {2 b \left (107 a^4+192 a^2 b^2+16 b^4\right ) \sinh (c+d x)}{d}+\frac {15}{2} a x \left (8 a^4+40 a^2 b^2+15 b^4\right )\right )\right )+\frac {9 a b \sinh (c+d x) (a+b \cosh (c+d x))^3}{4 d}\right )+\frac {b \sinh (c+d x) (a+b \cosh (c+d x))^4}{5 d}\) |
Input:
Int[(a + b*Cosh[c + d*x])^5,x]
Output:
(b*(a + b*Cosh[c + d*x])^4*Sinh[c + d*x])/(5*d) + ((9*a*b*(a + b*Cosh[c + d*x])^3*Sinh[c + d*x])/(4*d) + ((b*(47*a^2 + 16*b^2)*(a + b*Cosh[c + d*x]) ^2*Sinh[c + d*x])/(3*d) + ((15*a*(8*a^4 + 40*a^2*b^2 + 15*b^4)*x)/2 + (2*b *(107*a^4 + 192*a^2*b^2 + 16*b^4)*Sinh[c + d*x])/d + (7*a*b^2*(22*a^2 + 23 *b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d))/3)/4)/5
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 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]
Time = 149.97 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {600 \left (2 a^{3} b^{2}+a \,b^{4}\right ) \sinh \left (2 d x +2 c \right )+50 \left (8 a^{2} b^{3}+b^{5}\right ) \sinh \left (3 d x +3 c \right )+75 a \,b^{4} \sinh \left (4 d x +4 c \right )+6 b^{5} \sinh \left (5 d x +5 c \right )+300 \left (8 a^{4} b +12 a^{2} b^{3}+b^{5}\right ) \sinh \left (d x +c \right )+480 a d \left (a^{4}+5 a^{2} b^{2}+\frac {15}{8} b^{4}\right ) x}{480 d}\) | \(132\) |
| derivativedivides | \(\frac {b^{5} \left (\frac {8}{15}+\frac {\cosh \left (d x +c \right )^{4}}{5}+\frac {4 \cosh \left (d x +c \right )^{2}}{15}\right ) \sinh \left (d x +c \right )+5 a \,b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+10 a^{2} b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+10 a^{3} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+5 a^{4} b \sinh \left (d x +c \right )+a^{5} \left (d x +c \right )}{d}\) | \(155\) |
| default | \(\frac {b^{5} \left (\frac {8}{15}+\frac {\cosh \left (d x +c \right )^{4}}{5}+\frac {4 \cosh \left (d x +c \right )^{2}}{15}\right ) \sinh \left (d x +c \right )+5 a \,b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+10 a^{2} b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+10 a^{3} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+5 a^{4} b \sinh \left (d x +c \right )+a^{5} \left (d x +c \right )}{d}\) | \(155\) |
| parts | \(a^{5} x +\frac {b^{5} \left (\frac {8}{15}+\frac {\cosh \left (d x +c \right )^{4}}{5}+\frac {4 \cosh \left (d x +c \right )^{2}}{15}\right ) \sinh \left (d x +c \right )}{d}+\frac {5 a^{4} b \sinh \left (d x +c \right )}{d}+\frac {10 a^{3} b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )}{d}+\frac {5 a \,b^{4} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(162\) |
| risch | \(a^{5} x +5 a^{3} b^{2} x +\frac {15 a \,b^{4} x}{8}+\frac {b^{5} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {5 a \,b^{4} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {5 b^{3} {\mathrm e}^{3 d x +3 c} a^{2}}{12 d}+\frac {5 b^{5} {\mathrm e}^{3 d x +3 c}}{96 d}+\frac {5 b^{2} a^{3} {\mathrm e}^{2 d x +2 c}}{4 d}+\frac {5 b^{4} a \,{\mathrm e}^{2 d x +2 c}}{8 d}+\frac {5 b \,{\mathrm e}^{d x +c} a^{4}}{2 d}+\frac {15 b^{3} {\mathrm e}^{d x +c} a^{2}}{4 d}+\frac {5 b^{5} {\mathrm e}^{d x +c}}{16 d}-\frac {5 b \,{\mathrm e}^{-d x -c} a^{4}}{2 d}-\frac {15 b^{3} {\mathrm e}^{-d x -c} a^{2}}{4 d}-\frac {5 b^{5} {\mathrm e}^{-d x -c}}{16 d}-\frac {5 b^{2} a^{3} {\mathrm e}^{-2 d x -2 c}}{4 d}-\frac {5 b^{4} a \,{\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {5 b^{3} {\mathrm e}^{-3 d x -3 c} a^{2}}{12 d}-\frac {5 b^{5} {\mathrm e}^{-3 d x -3 c}}{96 d}-\frac {5 a \,b^{4} {\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {b^{5} {\mathrm e}^{-5 d x -5 c}}{160 d}\) | \(344\) |
| orering | \(\text {Expression too large to display}\) | \(1796\) |
Input:
int((a+b*cosh(d*x+c))^5,x,method=_RETURNVERBOSE)
Output:
1/480*(600*(2*a^3*b^2+a*b^4)*sinh(2*d*x+2*c)+50*(8*a^2*b^3+b^5)*sinh(3*d*x +3*c)+75*a*b^4*sinh(4*d*x+4*c)+6*b^5*sinh(5*d*x+5*c)+300*(8*a^4*b+12*a^2*b ^3+b^5)*sinh(d*x+c)+480*a*d*(a^4+5*a^2*b^2+15/8*b^4)*x)/d
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {3 \, b^{5} \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b^{5} \cosh \left (d x + c\right )^{2} + 30 \, a b^{4} \cosh \left (d x + c\right ) + 40 \, a^{2} b^{3} + 5 \, b^{5}\right )} \sinh \left (d x + c\right )^{3} + 30 \, {\left (8 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 15 \, {\left (b^{5} \cosh \left (d x + c\right )^{4} + 10 \, a b^{4} \cosh \left (d x + c\right )^{3} + 80 \, a^{4} b + 120 \, a^{2} b^{3} + 10 \, b^{5} + 5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (d x + c\right )^{2} + 40 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{240 \, d} \] Input:
integrate((a+b*cosh(d*x+c))^5,x, algorithm="fricas")
Output:
1/240*(3*b^5*sinh(d*x + c)^5 + 5*(6*b^5*cosh(d*x + c)^2 + 30*a*b^4*cosh(d* x + c) + 40*a^2*b^3 + 5*b^5)*sinh(d*x + c)^3 + 30*(8*a^5 + 40*a^3*b^2 + 15 *a*b^4)*d*x + 15*(b^5*cosh(d*x + c)^4 + 10*a*b^4*cosh(d*x + c)^3 + 80*a^4* b + 120*a^2*b^3 + 10*b^5 + 5*(8*a^2*b^3 + b^5)*cosh(d*x + c)^2 + 40*(2*a^3 *b^2 + a*b^4)*cosh(d*x + c))*sinh(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.72 \[ \int (a+b \cosh (c+d x))^5 \, dx=\begin {cases} a^{5} x + \frac {5 a^{4} b \sinh {\left (c + d x \right )}}{d} - 5 a^{3} b^{2} x \sinh ^{2}{\left (c + d x \right )} + 5 a^{3} b^{2} x \cosh ^{2}{\left (c + d x \right )} + \frac {5 a^{3} b^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {20 a^{2} b^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {10 a^{2} b^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} + \frac {15 a b^{4} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {15 a b^{4} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {15 a b^{4} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {15 a b^{4} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {25 a b^{4} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 b^{5} \sinh ^{5}{\left (c + d x \right )}}{15 d} - \frac {4 b^{5} \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} + \frac {b^{5} \sinh {\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cosh {\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*cosh(d*x+c))**5,x)
Output:
Piecewise((a**5*x + 5*a**4*b*sinh(c + d*x)/d - 5*a**3*b**2*x*sinh(c + d*x) **2 + 5*a**3*b**2*x*cosh(c + d*x)**2 + 5*a**3*b**2*sinh(c + d*x)*cosh(c + d*x)/d - 20*a**2*b**3*sinh(c + d*x)**3/(3*d) + 10*a**2*b**3*sinh(c + d*x)* cosh(c + d*x)**2/d + 15*a*b**4*x*sinh(c + d*x)**4/8 - 15*a*b**4*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 15*a*b**4*x*cosh(c + d*x)**4/8 - 15*a*b**4*s inh(c + d*x)**3*cosh(c + d*x)/(8*d) + 25*a*b**4*sinh(c + d*x)*cosh(c + d*x )**3/(8*d) + 8*b**5*sinh(c + d*x)**5/(15*d) - 4*b**5*sinh(c + d*x)**3*cosh (c + d*x)**2/(3*d) + b**5*sinh(c + d*x)*cosh(c + d*x)**4/d, Ne(d, 0)), (x* (a + b*cosh(c))**5, True))
Time = 0.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.49 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {5}{64} \, a b^{4} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {5}{4} \, a^{3} b^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{5} x + \frac {1}{480} \, b^{5} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} - \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {5}{12} \, a^{2} b^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {5 \, a^{4} b \sinh \left (d x + c\right )}{d} \] Input:
integrate((a+b*cosh(d*x+c))^5,x, algorithm="maxima")
Output:
5/64*a*b^4*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) + 5/4*a^3*b^2*(4*x + e^(2*d*x + 2*c)/d - e^( -2*d*x - 2*c)/d) + a^5*x + 1/480*b^5*(3*e^(5*d*x + 5*c)/d + 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d - 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d - 3*e^(-5*d*x - 5*c)/d) + 5/12*a^2*b^3*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d*x - 3*c)/d) + 5*a^4*b*sinh(d*x + c)/d
Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.44 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {b^{5} e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} + \frac {5 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} - \frac {5 \, a b^{4} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} - \frac {b^{5} e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} + \frac {1}{8} \, {\left (8 \, a^{5} + 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x + \frac {5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {5 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac {5 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + b^{5}\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {5 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {5 \, {\left (2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} - \frac {5 \, {\left (8 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} \] Input:
integrate((a+b*cosh(d*x+c))^5,x, algorithm="giac")
Output:
1/160*b^5*e^(5*d*x + 5*c)/d + 5/64*a*b^4*e^(4*d*x + 4*c)/d - 5/64*a*b^4*e^ (-4*d*x - 4*c)/d - 1/160*b^5*e^(-5*d*x - 5*c)/d + 1/8*(8*a^5 + 40*a^3*b^2 + 15*a*b^4)*x + 5/96*(8*a^2*b^3 + b^5)*e^(3*d*x + 3*c)/d + 5/8*(2*a^3*b^2 + a*b^4)*e^(2*d*x + 2*c)/d + 5/16*(8*a^4*b + 12*a^2*b^3 + b^5)*e^(d*x + c) /d - 5/16*(8*a^4*b + 12*a^2*b^3 + b^5)*e^(-d*x - c)/d - 5/8*(2*a^3*b^2 + a *b^4)*e^(-2*d*x - 2*c)/d - 5/96*(8*a^2*b^3 + b^5)*e^(-3*d*x - 3*c)/d
Time = 2.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.87 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {75\,b^5\,\mathrm {sinh}\left (c+d\,x\right )+\frac {25\,b^5\,\mathrm {sinh}\left (3\,c+3\,d\,x\right )}{2}+\frac {3\,b^5\,\mathrm {sinh}\left (5\,c+5\,d\,x\right )}{2}+150\,a\,b^4\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {75\,a\,b^4\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+900\,a^2\,b^3\,\mathrm {sinh}\left (c+d\,x\right )+300\,a^3\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+100\,a^2\,b^3\,\mathrm {sinh}\left (3\,c+3\,d\,x\right )+600\,a^4\,b\,\mathrm {sinh}\left (c+d\,x\right )+120\,a^5\,d\,x+225\,a\,b^4\,d\,x+600\,a^3\,b^2\,d\,x}{120\,d} \] Input:
int((a + b*cosh(c + d*x))^5,x)
Output:
(75*b^5*sinh(c + d*x) + (25*b^5*sinh(3*c + 3*d*x))/2 + (3*b^5*sinh(5*c + 5 *d*x))/2 + 150*a*b^4*sinh(2*c + 2*d*x) + (75*a*b^4*sinh(4*c + 4*d*x))/4 + 900*a^2*b^3*sinh(c + d*x) + 300*a^3*b^2*sinh(2*c + 2*d*x) + 100*a^2*b^3*si nh(3*c + 3*d*x) + 600*a^4*b*sinh(c + d*x) + 120*a^5*d*x + 225*a*b^4*d*x + 600*a^3*b^2*d*x)/(120*d)
Time = 0.22 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.93 \[ \int (a+b \cosh (c+d x))^5 \, dx=\frac {6 e^{10 d x +10 c} b^{5}+75 e^{9 d x +9 c} a \,b^{4}+400 e^{8 d x +8 c} a^{2} b^{3}+50 e^{8 d x +8 c} b^{5}+1200 e^{7 d x +7 c} a^{3} b^{2}+600 e^{7 d x +7 c} a \,b^{4}+2400 e^{6 d x +6 c} a^{4} b +3600 e^{6 d x +6 c} a^{2} b^{3}+300 e^{6 d x +6 c} b^{5}+960 e^{5 d x +5 c} a^{5} d x +4800 e^{5 d x +5 c} a^{3} b^{2} d x +1800 e^{5 d x +5 c} a \,b^{4} d x -2400 e^{4 d x +4 c} a^{4} b -3600 e^{4 d x +4 c} a^{2} b^{3}-300 e^{4 d x +4 c} b^{5}-1200 e^{3 d x +3 c} a^{3} b^{2}-600 e^{3 d x +3 c} a \,b^{4}-400 e^{2 d x +2 c} a^{2} b^{3}-50 e^{2 d x +2 c} b^{5}-75 e^{d x +c} a \,b^{4}-6 b^{5}}{960 e^{5 d x +5 c} d} \] Input:
int((a+b*cosh(d*x+c))^5,x)
Output:
(6*e**(10*c + 10*d*x)*b**5 + 75*e**(9*c + 9*d*x)*a*b**4 + 400*e**(8*c + 8* d*x)*a**2*b**3 + 50*e**(8*c + 8*d*x)*b**5 + 1200*e**(7*c + 7*d*x)*a**3*b** 2 + 600*e**(7*c + 7*d*x)*a*b**4 + 2400*e**(6*c + 6*d*x)*a**4*b + 3600*e**( 6*c + 6*d*x)*a**2*b**3 + 300*e**(6*c + 6*d*x)*b**5 + 960*e**(5*c + 5*d*x)* a**5*d*x + 4800*e**(5*c + 5*d*x)*a**3*b**2*d*x + 1800*e**(5*c + 5*d*x)*a*b **4*d*x - 2400*e**(4*c + 4*d*x)*a**4*b - 3600*e**(4*c + 4*d*x)*a**2*b**3 - 300*e**(4*c + 4*d*x)*b**5 - 1200*e**(3*c + 3*d*x)*a**3*b**2 - 600*e**(3*c + 3*d*x)*a*b**4 - 400*e**(2*c + 2*d*x)*a**2*b**3 - 50*e**(2*c + 2*d*x)*b* *5 - 75*e**(c + d*x)*a*b**4 - 6*b**5)/(960*e**(5*c + 5*d*x)*d)