Integrand size = 29, antiderivative size = 187 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {55 a^4 x}{256}-\frac {11 a^4 \cos ^7(c+d x)}{112 d}+\frac {55 a^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d} \]
55/256*a^4*x-11/112*a^4*cos(d*x+c)^7/d+55/256*a^4*cos(d*x+c)*sin(d*x+c)/d+ 55/384*a^4*cos(d*x+c)^3*sin(d*x+c)/d+11/96*a^4*cos(d*x+c)^5*sin(d*x+c)/d-1 /10*cos(d*x+c)^5*(a+a*sin(d*x+c))^5/a/d-1/18*cos(d*x+c)^7*(a^2+a^2*sin(d*x +c))^2/d-11/144*cos(d*x+c)^7*(a^4+a^4*sin(d*x+c))/d
Time = 0.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.62 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (136080 c+138600 d x-181440 \cos (c+d x)-53760 \cos (3 (c+d x))+16128 \cos (5 (c+d x))+7200 \cos (7 (c+d x))-1120 \cos (9 (c+d x))+8820 \sin (2 (c+d x))-42840 \sin (4 (c+d x))-2730 \sin (6 (c+d x))+4095 \sin (8 (c+d x))-126 \sin (10 (c+d x)))}{645120 d} \]
(a^4*(136080*c + 138600*d*x - 181440*Cos[c + d*x] - 53760*Cos[3*(c + d*x)] + 16128*Cos[5*(c + d*x)] + 7200*Cos[7*(c + d*x)] - 1120*Cos[9*(c + d*x)] + 8820*Sin[2*(c + d*x)] - 42840*Sin[4*(c + d*x)] - 2730*Sin[6*(c + d*x)] + 4095*Sin[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)
Time = 0.93 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3349, 3042, 3157, 3042, 3157, 3042, 3148, 3042, 3115, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^4 (a \sin (c+d x)+a)^4dx\) |
| \(\Big \downarrow \) 3349 |
| \(\displaystyle \frac {1}{2} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)^3dx-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)^3dx-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \int \cos ^6(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \int \cos (c+d x)^6 (\sin (c+d x) a+a)dx-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \int \cos ^6(c+d x)dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} a \left (\frac {11}{9} a \left (\frac {9}{8} a \left (a \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {a \cos ^7(c+d x)}{7 d}\right )-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}\right )-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d}\) |
-1/10*(Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(a*d) + (a*(-1/9*(a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/d + (11*a*(-1/8*(Cos[c + d*x]^7*(a^2 + a^2* Sin[c + d*x]))/d + (9*a*(-1/7*(a*Cos[c + d*x]^7)/d + a*((Cos[c + d*x]^5*Si n[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (C os[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)))/8))/9))/2
3.5.7.3.1 Defintions of rubi rules used
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^( p + 1))*((a + b*Sin[e + f*x])^(m + 1)/(2*b*f*g*(m + 1))), x] + Simp[a/(2*g^ 2) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; F reeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]
Time = 0.97 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(-\frac {a^{4} \left (-138600 d x +181440 \cos \left (d x +c \right )+126 \sin \left (10 d x +10 c \right )+1120 \cos \left (9 d x +9 c \right )-4095 \sin \left (8 d x +8 c \right )-7200 \cos \left (7 d x +7 c \right )+2730 \sin \left (6 d x +6 c \right )-16128 \cos \left (5 d x +5 c \right )+42840 \sin \left (4 d x +4 c \right )-8820 \sin \left (2 d x +2 c \right )+53760 \cos \left (3 d x +3 c \right )+212992\right )}{645120 d}\) | \(122\) |
| risch | \(\frac {55 a^{4} x}{256}-\frac {9 a^{4} \cos \left (d x +c \right )}{32 d}-\frac {a^{4} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{4} \cos \left (9 d x +9 c \right )}{576 d}+\frac {13 a^{4} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {5 a^{4} \cos \left (7 d x +7 c \right )}{448 d}-\frac {13 a^{4} \sin \left (6 d x +6 c \right )}{3072 d}+\frac {a^{4} \cos \left (5 d x +5 c \right )}{40 d}-\frac {17 a^{4} \sin \left (4 d x +4 c \right )}{256 d}-\frac {a^{4} \cos \left (3 d x +3 c \right )}{12 d}+\frac {7 a^{4} \sin \left (2 d x +2 c \right )}{512 d}\) | \(175\) |
| derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+4 a^{4} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+6 a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+4 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(306\) |
| default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+4 a^{4} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+6 a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+4 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(306\) |
-1/645120*a^4*(-138600*d*x+181440*cos(d*x+c)+126*sin(10*d*x+10*c)+1120*cos (9*d*x+9*c)-4095*sin(8*d*x+8*c)-7200*cos(7*d*x+7*c)+2730*sin(6*d*x+6*c)-16 128*cos(5*d*x+5*c)+42840*sin(4*d*x+4*c)-8820*sin(2*d*x+2*c)+53760*cos(3*d* x+3*c)+212992)/d
Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {35840 \, a^{4} \cos \left (d x + c\right )^{9} - 138240 \, a^{4} \cos \left (d x + c\right )^{7} + 129024 \, a^{4} \cos \left (d x + c\right )^{5} - 17325 \, a^{4} d x + 21 \, {\left (384 \, a^{4} \cos \left (d x + c\right )^{9} - 3888 \, a^{4} \cos \left (d x + c\right )^{7} + 5704 \, a^{4} \cos \left (d x + c\right )^{5} - 550 \, a^{4} \cos \left (d x + c\right )^{3} - 825 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
-1/80640*(35840*a^4*cos(d*x + c)^9 - 138240*a^4*cos(d*x + c)^7 + 129024*a^ 4*cos(d*x + c)^5 - 17325*a^4*d*x + 21*(384*a^4*cos(d*x + c)^9 - 3888*a^4*c os(d*x + c)^7 + 5704*a^4*cos(d*x + c)^5 - 550*a^4*cos(d*x + c)^3 - 825*a^4 *cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (173) = 346\).
Time = 1.31 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.99 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\begin {cases} \frac {3 a^{4} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{4} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {9 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {27 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {9 a^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} - \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {33 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {4 a^{4} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {33 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} + \frac {a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {16 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {9 a^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {32 a^{4} \cos ^{9}{\left (c + d x \right )}}{315 d} - \frac {8 a^{4} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{4} \sin ^{2}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a**4*x*sin(c + d*x)**10/256 + 15*a**4*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 9*a**4*x*sin(c + d*x)**8/64 + 15*a**4*x*sin(c + d*x)**6*c os(c + d*x)**4/128 + 9*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + a**4*x* sin(c + d*x)**6/16 + 15*a**4*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 27*a* *4*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 3*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**4*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 9*a**4*x*sin (c + d*x)**2*cos(c + d*x)**6/16 + 3*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4 /16 + 3*a**4*x*cos(c + d*x)**10/256 + 9*a**4*x*cos(c + d*x)**8/64 + a**4*x *cos(c + d*x)**6/16 + 3*a**4*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**4 *sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 9*a**4*sin(c + d*x)**7*cos(c + d*x)/(64*d) - a**4*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 33*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(64*d) + a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d ) - 4*a**4*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**4*sin(c + d*x)**3* cos(c + d*x)**7/(128*d) - 33*a**4*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) + a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 16*a**4*sin(c + d*x)**2*cos( c + d*x)**7/(35*d) - 4*a**4*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**4 *sin(c + d*x)*cos(c + d*x)**9/(256*d) - 9*a**4*sin(c + d*x)*cos(c + d*x)** 7/(64*d) - a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 32*a**4*cos(c + d*x) **9/(315*d) - 8*a**4*cos(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a*sin(c) + a)* *4*sin(c)**2*cos(c)**4, True))
Time = 0.22 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.99 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {8192 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{4} - 73728 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 3780 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{645120 \, d} \]
-1/645120*(8192*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5 )*a^4 - 73728*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^4 + 63*(32*sin(2*d*x + 2*c)^5 - 120*d*x - 120*c - 5*sin(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*a^ 4 - 3360*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^4 - 3780*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^4)/d
Time = 0.62 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.93 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {55}{256} \, a^{4} x - \frac {a^{4} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} + \frac {5 \, a^{4} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a^{4} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {a^{4} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {9 \, a^{4} \cos \left (d x + c\right )}{32 \, d} - \frac {a^{4} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {13 \, a^{4} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {13 \, a^{4} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {17 \, a^{4} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
55/256*a^4*x - 1/576*a^4*cos(9*d*x + 9*c)/d + 5/448*a^4*cos(7*d*x + 7*c)/d + 1/40*a^4*cos(5*d*x + 5*c)/d - 1/12*a^4*cos(3*d*x + 3*c)/d - 9/32*a^4*co s(d*x + c)/d - 1/5120*a^4*sin(10*d*x + 10*c)/d + 13/2048*a^4*sin(8*d*x + 8 *c)/d - 13/3072*a^4*sin(6*d*x + 6*c)/d - 17/256*a^4*sin(4*d*x + 4*c)/d + 7 /512*a^4*sin(2*d*x + 2*c)/d
Time = 12.76 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.06 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {55\,a^4\,x}{256}-\frac {\frac {571\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384}-\frac {14149\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}-\frac {469\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {4293\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {4293\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {469\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {14149\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{480}-\frac {571\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{384}-\frac {55\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{80640}-\frac {a^4\,\left (17325\,c+17325\,d\,x-53248\right )}{80640}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{8064}-\frac {a^4\,\left (173250\,c+173250\,d\,x\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{8064}-\frac {a^4\,\left (173250\,c+173250\,d\,x-532480\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{1792}-\frac {a^4\,\left (779625\,c+779625\,d\,x-1105920\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{1792}-\frac {a^4\,\left (779625\,c+779625\,d\,x-1290240\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{672}-\frac {a^4\,\left (2079000\,c+2079000\,d\,x-368640\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{384}-\frac {a^4\,\left (3638250\,c+3638250\,d\,x-860160\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{672}-\frac {a^4\,\left (2079000\,c+2079000\,d\,x-6021120\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{320}-\frac {a^4\,\left (4365900\,c+4365900\,d\,x-6709248\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{384}-\frac {a^4\,\left (3638250\,c+3638250\,d\,x-10321920\right )}{80640}\right )+\frac {55\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
(55*a^4*x)/256 - ((571*a^4*tan(c/2 + (d*x)/2)^3)/384 - (14149*a^4*tan(c/2 + (d*x)/2)^5)/480 - (469*a^4*tan(c/2 + (d*x)/2)^7)/32 + (4293*a^4*tan(c/2 + (d*x)/2)^9)/64 - (4293*a^4*tan(c/2 + (d*x)/2)^11)/64 + (469*a^4*tan(c/2 + (d*x)/2)^13)/32 + (14149*a^4*tan(c/2 + (d*x)/2)^15)/480 - (571*a^4*tan(c /2 + (d*x)/2)^17)/384 - (55*a^4*tan(c/2 + (d*x)/2)^19)/128 + (a^4*(17325*c + 17325*d*x))/80640 - (a^4*(17325*c + 17325*d*x - 53248))/80640 + tan(c/2 + (d*x)/2)^18*((a^4*(17325*c + 17325*d*x))/8064 - (a^4*(173250*c + 173250 *d*x))/80640) + tan(c/2 + (d*x)/2)^2*((a^4*(17325*c + 17325*d*x))/8064 - ( a^4*(173250*c + 173250*d*x - 532480))/80640) + tan(c/2 + (d*x)/2)^4*((a^4* (17325*c + 17325*d*x))/1792 - (a^4*(779625*c + 779625*d*x - 1105920))/8064 0) + tan(c/2 + (d*x)/2)^16*((a^4*(17325*c + 17325*d*x))/1792 - (a^4*(77962 5*c + 779625*d*x - 1290240))/80640) + tan(c/2 + (d*x)/2)^6*((a^4*(17325*c + 17325*d*x))/672 - (a^4*(2079000*c + 2079000*d*x - 368640))/80640) + tan( c/2 + (d*x)/2)^12*((a^4*(17325*c + 17325*d*x))/384 - (a^4*(3638250*c + 363 8250*d*x - 860160))/80640) + tan(c/2 + (d*x)/2)^14*((a^4*(17325*c + 17325* d*x))/672 - (a^4*(2079000*c + 2079000*d*x - 6021120))/80640) + tan(c/2 + ( d*x)/2)^10*((a^4*(17325*c + 17325*d*x))/320 - (a^4*(4365900*c + 4365900*d* x - 6709248))/80640) + tan(c/2 + (d*x)/2)^8*((a^4*(17325*c + 17325*d*x))/3 84 - (a^4*(3638250*c + 3638250*d*x - 10321920))/80640) + (55*a^4*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)