Integrand size = 27, antiderivative size = 143 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{128}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^5(c+d x) \sin ^3(c+d x)}{8 d} \] Output:
3/128*a*x-1/5*a*cos(d*x+c)^5/d+2/7*a*cos(d*x+c)^7/d-1/9*a*cos(d*x+c)^9/d+3 /128*a*cos(d*x+c)*sin(d*x+c)/d+1/64*a*cos(d*x+c)^3*sin(d*x+c)/d-1/16*a*cos (d*x+c)^5*sin(d*x+c)/d-1/8*a*cos(d*x+c)^5*sin(d*x+c)^3/d
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.59 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (7560 c+7560 d x-7560 \cos (c+d x)-1680 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+180 \cos (7 (c+d x))-140 \cos (9 (c+d x))-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x)))}{322560 d} \] Input:
Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]
Output:
(a*(7560*c + 7560*d*x - 7560*Cos[c + d*x] - 1680*Cos[3*(c + d*x)] + 1008*C os[5*(c + d*x)] + 180*Cos[7*(c + d*x)] - 140*Cos[9*(c + d*x)] - 2520*Sin[4 *(c + d*x)] + 315*Sin[8*(c + d*x)]))/(322560*d)
Time = 0.71 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3317, 3042, 3045, 244, 2009, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^4(c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^4 \cos (c+d x)^4 (a \sin (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^4(c+d x) \sin ^5(c+d x)dx+a \int \cos ^4(c+d x) \sin ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \cos (c+d x)^4 \sin (c+d x)^4dx+a \int \cos (c+d x)^4 \sin (c+d x)^5dx\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle a \int \cos (c+d x)^4 \sin (c+d x)^4dx-\frac {a \int \cos ^4(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle a \int \cos (c+d x)^4 \sin (c+d x)^4dx-\frac {a \int \left (\cos ^8(c+d x)-2 \cos ^6(c+d x)+\cos ^4(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \cos (c+d x)^4 \sin (c+d x)^4dx-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {3}{8} \int \cos ^4(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{8} \int \cos (c+d x)^4 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \int \cos ^4(c+d x)dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin ^3(c+d x) \cos ^5(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{9} \cos ^9(c+d x)-\frac {2}{7} \cos ^7(c+d x)+\frac {1}{5} \cos ^5(c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^5/5 - (2*Cos[c + d*x]^7)/7 + Cos[c + d*x]^9/9))/d) + a* (-1/8*(Cos[c + d*x]^5*Sin[c + d*x]^3)/d + (3*(-1/6*(Cos[c + d*x]^5*Sin[c + d*x])/d + ((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]* Sin[c + d*x])/(2*d)))/4)/6))/8)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 79.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(-\frac {\left (-3 d x +\sin \left (4 d x +4 c \right )-\frac {\sin \left (8 d x +8 c \right )}{8}+3 \cos \left (d x +c \right )+\frac {2 \cos \left (3 d x +3 c \right )}{3}-\frac {2 \cos \left (5 d x +5 c \right )}{5}-\frac {\cos \left (7 d x +7 c \right )}{14}+\frac {\cos \left (9 d x +9 c \right )}{18}+\frac {1024}{315}\right ) a}{128 d}\) | \(85\) |
| risch | \(\frac {3 a x}{128}-\frac {3 a \cos \left (d x +c \right )}{128 d}-\frac {a \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a \cos \left (7 d x +7 c \right )}{1792 d}+\frac {a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {a \cos \left (3 d x +3 c \right )}{192 d}\) | \(108\) |
| derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )+a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(124\) |
| default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )+a \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\) | \(124\) |
| norman | \(\frac {\frac {27 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32}-\frac {16 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d}-\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {16 a}{315 d}+\frac {27 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128}-\frac {13 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32 d}+\frac {155 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 d}-\frac {64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d}+\frac {63 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32}+\frac {3 a x}{128}+\frac {169 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{32 d}-\frac {155 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{32 d}-\frac {112 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d}+\frac {13 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{32 d}+\frac {16 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {32 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}-\frac {169 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 d}+\frac {189 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64}+\frac {32 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5 d}+\frac {189 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64}+\frac {3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{64 d}+\frac {27 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{32}+\frac {27 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128}+\frac {3 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}}{128}+\frac {63 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(399\) |
| orering | \(\text {Expression too large to display}\) | \(3041\) |
Input:
int(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/128*(-3*d*x+sin(4*d*x+4*c)-1/8*sin(8*d*x+8*c)+3*cos(d*x+c)+2/3*cos(3*d* x+3*c)-2/5*cos(5*d*x+5*c)-1/14*cos(7*d*x+7*c)+1/18*cos(9*d*x+9*c)+1024/315 )*a/d
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4480 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 8064 \, a \cos \left (d x + c\right )^{5} - 945 \, a d x - 315 \, {\left (16 \, a \cos \left (d x + c\right )^{7} - 24 \, a \cos \left (d x + c\right )^{5} + 2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/40320*(4480*a*cos(d*x + c)^9 - 11520*a*cos(d*x + c)^7 + 8064*a*cos(d*x + c)^5 - 945*a*d*x - 315*(16*a*cos(d*x + c)^7 - 24*a*cos(d*x + c)^5 + 2*a* cos(d*x + c)^3 + 3*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (131) = 262\).
Time = 0.98 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.90 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {11 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {4 a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**4*(a+a*sin(d*x+c)),x)
Output:
Piecewise((3*a*x*sin(c + d*x)**8/128 + 3*a*x*sin(c + d*x)**6*cos(c + d*x)* *2/32 + 9*a*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 3*a*x*sin(c + d*x)**2*c os(c + d*x)**6/32 + 3*a*x*cos(c + d*x)**8/128 + 3*a*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*a*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - a*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 11*a*sin(c + d*x)**3*cos(c + d*x)**5/(128* d) - 4*a*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 3*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 8*a*cos(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**4*cos(c)**4, True))
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.50 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a - 315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{322560 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/322560*(1024*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5 )*a - 315*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a)/d
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{128} \, a x - \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
3/128*a*x - 1/2304*a*cos(9*d*x + 9*c)/d + 1/1792*a*cos(7*d*x + 7*c)/d + 1/ 320*a*cos(5*d*x + 5*c)/d - 1/192*a*cos(3*d*x + 3*c)/d - 3/128*a*cos(d*x + c)/d + 1/1024*a*sin(8*d*x + 8*c)/d - 1/128*a*sin(4*d*x + 4*c)/d
Time = 20.65 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.47 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{128}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}-\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (79380\,c+79380\,d\,x-430080\right )}{40320}-\frac {63\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\left (\frac {a\,\left (119070\,c+119070\,d\,x+645120\right )}{40320}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {a\,\left (119070\,c+119070\,d\,x-903168\right )}{40320}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {169\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (79380\,c+79380\,d\,x+258048\right )}{40320}-\frac {63\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {155\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\left (\frac {a\,\left (34020\,c+34020\,d\,x-73728\right )}{40320}-\frac {27\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}+\left (\frac {a\,\left (8505\,c+8505\,d\,x-18432\right )}{40320}-\frac {27\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (945\,c+945\,d\,x-2048\right )}{40320}-\frac {3\,a\,\left (c+d\,x\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \] Input:
int(cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x)),x)
Output:
(3*a*x)/128 + ((a*(945*c + 945*d*x - 2048))/40320 - (3*a*tan(c/2 + (d*x)/2 ))/64 - (3*a*(c + d*x))/128 + tan(c/2 + (d*x)/2)^2*((a*(8505*c + 8505*d*x - 18432))/40320 - (27*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^4*((a*(34020* c + 34020*d*x - 73728))/40320 - (27*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^ 6*((a*(79380*c + 79380*d*x + 258048))/40320 - (63*a*(c + d*x))/32) + tan(c /2 + (d*x)/2)^12*((a*(79380*c + 79380*d*x - 430080))/40320 - (63*a*(c + d* x))/32) + tan(c/2 + (d*x)/2)^10*((a*(119070*c + 119070*d*x + 645120))/4032 0 - (189*a*(c + d*x))/64) + tan(c/2 + (d*x)/2)^8*((a*(119070*c + 119070*d* x - 903168))/40320 - (189*a*(c + d*x))/64) - (13*a*tan(c/2 + (d*x)/2)^3)/3 2 + (155*a*tan(c/2 + (d*x)/2)^5)/32 - (169*a*tan(c/2 + (d*x)/2)^7)/32 + (1 69*a*tan(c/2 + (d*x)/2)^11)/32 - (155*a*tan(c/2 + (d*x)/2)^13)/32 + (13*a* tan(c/2 + (d*x)/2)^15)/32 + (3*a*tan(c/2 + (d*x)/2)^17)/64)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^9)
Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.02 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-5040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}+6400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+7560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-512 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-945 \cos \left (d x +c \right ) \sin \left (d x +c \right )-1024 \cos \left (d x +c \right )+945 d x +1024\right )}{40320 d} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c)),x)
Output:
(a*( - 4480*cos(c + d*x)*sin(c + d*x)**8 - 5040*cos(c + d*x)*sin(c + d*x)* *7 + 6400*cos(c + d*x)*sin(c + d*x)**6 + 7560*cos(c + d*x)*sin(c + d*x)**5 - 384*cos(c + d*x)*sin(c + d*x)**4 - 630*cos(c + d*x)*sin(c + d*x)**3 - 5 12*cos(c + d*x)*sin(c + d*x)**2 - 945*cos(c + d*x)*sin(c + d*x) - 1024*cos (c + d*x) + 945*d*x + 1024))/(40320*d)