Integrand size = 24, antiderivative size = 157 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=4 a \left (5 a^2+3 c^2\right ) x-\frac {4 c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{3 e}+\frac {4 a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{3 e}-\frac {20 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a+a \cos (d+e x)+c \sin (d+e x))}{3 e}-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a+a \cos (d+e x)+c \sin (d+e x))^2}{3 e} \]
4*a*(5*a^2+3*c^2)*x-4/3*c*(15*a^2+4*c^2)*cos(e*x+d)/e+4/3*a*(15*a^2+4*c^2) *sin(e*x+d)/e-20/3*(a*c*cos(e*x+d)-a^2*sin(e*x+d))*(a+a*cos(e*x+d)+c*sin(e *x+d))/e-8/3*(c*cos(e*x+d)-a*sin(e*x+d))*(a+a*cos(e*x+d)+c*sin(e*x+d))^2/e
Time = 1.46 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.86 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=\frac {2 \left (6 a \left (5 a^2+3 c^2\right ) (d+e x)-9 c \left (5 a^2+c^2\right ) \cos (d+e x)-18 a^2 c \cos (2 (d+e x))+c \left (-3 a^2+c^2\right ) \cos (3 (d+e x))+9 a \left (5 a^2+c^2\right ) \sin (d+e x)+9 a \left (a^2-c^2\right ) \sin (2 (d+e x))+a \left (a^2-3 c^2\right ) \sin (3 (d+e x))\right )}{3 e} \]
(2*(6*a*(5*a^2 + 3*c^2)*(d + e*x) - 9*c*(5*a^2 + c^2)*Cos[d + e*x] - 18*a^ 2*c*Cos[2*(d + e*x)] + c*(-3*a^2 + c^2)*Cos[3*(d + e*x)] + 9*a*(5*a^2 + c^ 2)*Sin[d + e*x] + 9*a*(a^2 - c^2)*Sin[2*(d + e*x)] + a*(a^2 - 3*c^2)*Sin[3 *(d + e*x)]))/(3*e)
Time = 0.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3599, 27, 3042, 3625, 2009}
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.
| \(\displaystyle \int (2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (2 a \cos (d+e x)+2 a+2 c \sin (d+e x))^3dx\) |
| \(\Big \downarrow \) 3599 |
| \(\displaystyle \frac {1}{3} \int 8 (\cos (d+e x) a+a+c \sin (d+e x)) \left (5 \cos (d+e x) a^2+5 a^2+5 c \sin (d+e x) a+2 c^2\right )dx-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8}{3} \int (\cos (d+e x) a+a+c \sin (d+e x)) \left (5 \cos (d+e x) a^2+5 a^2+5 c \sin (d+e x) a+2 c^2\right )dx-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{3} \int (\cos (d+e x) a+a+c \sin (d+e x)) \left (5 \cos (d+e x) a^2+5 a^2+5 c \sin (d+e x) a+2 c^2\right )dx-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {8}{3} \left (\frac {\int \left (3 \left (5 a^2+3 c^2\right ) a^2+\left (15 a^2+4 c^2\right ) \cos (d+e x) a^2+c \left (15 a^2+4 c^2\right ) \sin (d+e x) a\right )dx}{2 a}-\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a \cos (d+e x)+a+c \sin (d+e x))}{2 e}\right )-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8}{3} \left (\frac {\frac {a^2 \left (15 a^2+4 c^2\right ) \sin (d+e x)}{e}-\frac {a c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{e}+3 a^2 x \left (5 a^2+3 c^2\right )}{2 a}-\frac {5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right ) (a \cos (d+e x)+a+c \sin (d+e x))}{2 e}\right )-\frac {8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}\) |
(-8*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x] )^2)/(3*e) + (8*((-5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))/(2*e) + (3*a^2*(5*a^2 + 3*c^2)*x - (a*c*(15*a^2 + 4*c^2)*Cos[d + e*x])/e + (a^2*(15*a^2 + 4*c^2)*Sin[d + e*x])/e)/(2*a)))/3
3.4.63.3.1 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[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[(-(c*Cos[d + e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Simp[1/n Int[Simp[n*a^2 + ( n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x ], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 )) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
Time = 2.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\frac {\left (-6 a^{2} c +2 c^{3}\right ) \cos \left (3 e x +3 d \right )+\left (18 a^{3}-18 a \,c^{2}\right ) \sin \left (2 e x +2 d \right )+\left (2 a^{3}-6 a \,c^{2}\right ) \sin \left (3 e x +3 d \right )-36 a^{2} c \cos \left (2 e x +2 d \right )+\left (-90 a^{2} c -18 c^{3}\right ) \cos \left (e x +d \right )+\left (90 a^{3}+18 a \,c^{2}\right ) \sin \left (e x +d \right )+60 a^{3} e x +36 a \,c^{2} e x -60 a^{2} c -16 c^{3}}{3 e}\) | \(152\) |
| parts | \(\frac {-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}+\frac {8 a \left (a +c \sin \left (e x +d \right )\right )^{3}}{e c}+8 a^{3} x +\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3 e}-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3 e}-\frac {24 c \cos \left (e x +d \right ) a^{2}}{e}+\frac {24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )}{e}\) | \(169\) |
| derivativedivides | \(\frac {\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a \,c^{2} \sin \left (e x +d \right )^{3}-24 a^{2} c \cos \left (e x +d \right )^{2}+24 a^{3} \sin \left (e x +d \right )-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-24 a^{2} c \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(177\) |
| default | \(\frac {\frac {8 a^{3} \left (2+\cos \left (e x +d \right )^{2}\right ) \sin \left (e x +d \right )}{3}-8 a^{2} c \cos \left (e x +d \right )^{3}+24 a^{3} \left (\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )+8 a \,c^{2} \sin \left (e x +d \right )^{3}-24 a^{2} c \cos \left (e x +d \right )^{2}+24 a^{3} \sin \left (e x +d \right )-\frac {8 c^{3} \left (2+\sin \left (e x +d \right )^{2}\right ) \cos \left (e x +d \right )}{3}+24 a \,c^{2} \left (-\frac {\cos \left (e x +d \right ) \sin \left (e x +d \right )}{2}+\frac {e x}{2}+\frac {d}{2}\right )-24 a^{2} c \cos \left (e x +d \right )+8 a^{3} \left (e x +d \right )}{e}\) | \(177\) |
| risch | \(20 a^{3} x +12 a \,c^{2} x -\frac {30 c \cos \left (e x +d \right ) a^{2}}{e}-\frac {6 c^{3} \cos \left (e x +d \right )}{e}+\frac {30 a^{3} \sin \left (e x +d \right )}{e}+\frac {6 a \sin \left (e x +d \right ) c^{2}}{e}-\frac {2 c \cos \left (3 e x +3 d \right ) a^{2}}{e}+\frac {2 c^{3} \cos \left (3 e x +3 d \right )}{3 e}+\frac {2 a^{3} \sin \left (3 e x +3 d \right )}{3 e}-\frac {2 a \sin \left (3 e x +3 d \right ) c^{2}}{e}-\frac {12 a^{2} c \cos \left (2 e x +2 d \right )}{e}+\frac {6 a^{3} \sin \left (2 e x +2 d \right )}{e}-\frac {6 a \sin \left (2 e x +2 d \right ) c^{2}}{e}\) | \(196\) |
| norman | \(\frac {-\frac {192 a^{2} c +32 c^{3}}{3 e}+4 a \left (5 a^{2}+3 c^{2}\right ) x -\frac {32 c^{3} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{e}+\frac {64 a \left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 e}+\frac {8 a \left (5 a^{2}+3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{e}+12 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}+12 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}+4 a \left (5 a^{2}+3 c^{2}\right ) x \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}+\frac {8 a \left (11 a^{2}-3 c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{e}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(229\) |
1/3*((-6*a^2*c+2*c^3)*cos(3*e*x+3*d)+(18*a^3-18*a*c^2)*sin(2*e*x+2*d)+(2*a ^3-6*a*c^2)*sin(3*e*x+3*d)-36*a^2*c*cos(2*e*x+2*d)+(-90*a^2*c-18*c^3)*cos( e*x+d)+(90*a^3+18*a*c^2)*sin(e*x+d)+60*a^3*e*x+36*a*c^2*e*x-60*a^2*c-16*c^ 3)/e
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {4 \, {\left (18 \, a^{2} c \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} e x + 6 \, {\left (3 \, a^{2} c + c^{3}\right )} \cos \left (e x + d\right ) - {\left (22 \, a^{3} + 6 \, a c^{2} + 2 \, {\left (a^{3} - 3 \, a c^{2}\right )} \cos \left (e x + d\right )^{2} + 9 \, {\left (a^{3} - a c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )}}{3 \, e} \]
-4/3*(18*a^2*c*cos(e*x + d)^2 + 2*(3*a^2*c - c^3)*cos(e*x + d)^3 - 3*(5*a^ 3 + 3*a*c^2)*e*x + 6*(3*a^2*c + c^3)*cos(e*x + d) - (22*a^3 + 6*a*c^2 + 2* (a^3 - 3*a*c^2)*cos(e*x + d)^2 + 9*(a^3 - a*c^2)*cos(e*x + d))*sin(e*x + d ))/e
Time = 0.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.85 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=\begin {cases} 12 a^{3} x \sin ^{2}{\left (d + e x \right )} + 12 a^{3} x \cos ^{2}{\left (d + e x \right )} + 8 a^{3} x + \frac {16 a^{3} \sin ^{3}{\left (d + e x \right )}}{3 e} + \frac {8 a^{3} \sin {\left (d + e x \right )} \cos ^{2}{\left (d + e x \right )}}{e} + \frac {12 a^{3} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} + \frac {24 a^{3} \sin {\left (d + e x \right )}}{e} + \frac {24 a^{2} c \sin ^{2}{\left (d + e x \right )}}{e} - \frac {8 a^{2} c \cos ^{3}{\left (d + e x \right )}}{e} - \frac {24 a^{2} c \cos {\left (d + e x \right )}}{e} + 12 a c^{2} x \sin ^{2}{\left (d + e x \right )} + 12 a c^{2} x \cos ^{2}{\left (d + e x \right )} + \frac {8 a c^{2} \sin ^{3}{\left (d + e x \right )}}{e} - \frac {12 a c^{2} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {8 c^{3} \sin ^{2}{\left (d + e x \right )} \cos {\left (d + e x \right )}}{e} - \frac {16 c^{3} \cos ^{3}{\left (d + e x \right )}}{3 e} & \text {for}\: e \neq 0 \\x \left (2 a \cos {\left (d \right )} + 2 a + 2 c \sin {\left (d \right )}\right )^{3} & \text {otherwise} \end {cases} \]
Piecewise((12*a**3*x*sin(d + e*x)**2 + 12*a**3*x*cos(d + e*x)**2 + 8*a**3* x + 16*a**3*sin(d + e*x)**3/(3*e) + 8*a**3*sin(d + e*x)*cos(d + e*x)**2/e + 12*a**3*sin(d + e*x)*cos(d + e*x)/e + 24*a**3*sin(d + e*x)/e + 24*a**2*c *sin(d + e*x)**2/e - 8*a**2*c*cos(d + e*x)**3/e - 24*a**2*c*cos(d + e*x)/e + 12*a*c**2*x*sin(d + e*x)**2 + 12*a*c**2*x*cos(d + e*x)**2 + 8*a*c**2*si n(d + e*x)**3/e - 12*a*c**2*sin(d + e*x)*cos(d + e*x)/e - 8*c**3*sin(d + e *x)**2*cos(d + e*x)/e - 16*c**3*cos(d + e*x)**3/(3*e), Ne(e, 0)), (x*(2*a* cos(d) + 2*a + 2*c*sin(d))**3, True))
Time = 0.23 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {8 \, a^{2} c \cos \left (e x + d\right )^{3}}{e} + \frac {8 \, a c^{2} \sin \left (e x + d\right )^{3}}{e} + 8 \, a^{3} x - \frac {8 \, {\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} a^{3}}{3 \, e} + \frac {8 \, {\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{3 \, e} - 24 \, a^{2} {\left (\frac {c \cos \left (e x + d\right )}{e} - \frac {a \sin \left (e x + d\right )}{e}\right )} - 6 \, {\left (\frac {4 \, a c \cos \left (e x + d\right )^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} a^{2}}{e} - \frac {{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} a \]
-8*a^2*c*cos(e*x + d)^3/e + 8*a*c^2*sin(e*x + d)^3/e + 8*a^3*x - 8/3*(sin( e*x + d)^3 - 3*sin(e*x + d))*a^3/e + 8/3*(cos(e*x + d)^3 - 3*cos(e*x + d)) *c^3/e - 24*a^2*(c*cos(e*x + d)/e - a*sin(e*x + d)/e) - 6*(4*a*c*cos(e*x + d)^2/e - (2*e*x + 2*d + sin(2*e*x + 2*d))*a^2/e - (2*e*x + 2*d - sin(2*e* x + 2*d))*c^2/e)*a
Time = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=-\frac {12 \, a^{2} c \cos \left (2 \, e x + 2 \, d\right )}{e} + 4 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} x - \frac {2 \, {\left (3 \, a^{2} c - c^{3}\right )} \cos \left (3 \, e x + 3 \, d\right )}{3 \, e} - \frac {6 \, {\left (5 \, a^{2} c + c^{3}\right )} \cos \left (e x + d\right )}{e} + \frac {2 \, {\left (a^{3} - 3 \, a c^{2}\right )} \sin \left (3 \, e x + 3 \, d\right )}{3 \, e} + \frac {6 \, {\left (a^{3} - a c^{2}\right )} \sin \left (2 \, e x + 2 \, d\right )}{e} + \frac {6 \, {\left (5 \, a^{3} + a c^{2}\right )} \sin \left (e x + d\right )}{e} \]
-12*a^2*c*cos(2*e*x + 2*d)/e + 4*(5*a^3 + 3*a*c^2)*x - 2/3*(3*a^2*c - c^3) *cos(3*e*x + 3*d)/e - 6*(5*a^2*c + c^3)*cos(e*x + d)/e + 2/3*(a^3 - 3*a*c^ 2)*sin(3*e*x + 3*d)/e + 6*(a^3 - a*c^2)*sin(2*e*x + 2*d)/e + 6*(5*a^3 + a* c^2)*sin(e*x + d)/e
Time = 27.66 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.52 \[ \int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx=20\,a^3\,x-\frac {32\,c^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4}{e}+\frac {64\,c^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6}{3\,e}+12\,a\,c^2\,x-\frac {64\,a^2\,c\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6}{e}+\frac {40\,a^3\,\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}+\frac {80\,a^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{3\,e}+\frac {64\,a^3\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{3\,e}+\frac {16\,a\,c^2\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}-\frac {64\,a\,c^2\,{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e}+\frac {24\,a\,c^2\,\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e} \]
20*a^3*x - (32*c^3*cos(d/2 + (e*x)/2)^4)/e + (64*c^3*cos(d/2 + (e*x)/2)^6) /(3*e) + 12*a*c^2*x - (64*a^2*c*cos(d/2 + (e*x)/2)^6)/e + (40*a^3*cos(d/2 + (e*x)/2)*sin(d/2 + (e*x)/2))/e + (80*a^3*cos(d/2 + (e*x)/2)^3*sin(d/2 + (e*x)/2))/(3*e) + (64*a^3*cos(d/2 + (e*x)/2)^5*sin(d/2 + (e*x)/2))/(3*e) + (16*a*c^2*cos(d/2 + (e*x)/2)^3*sin(d/2 + (e*x)/2))/e - (64*a*c^2*cos(d/2 + (e*x)/2)^5*sin(d/2 + (e*x)/2))/e + (24*a*c^2*cos(d/2 + (e*x)/2)*sin(d/2 + (e*x)/2))/e