Integrand size = 27, antiderivative size = 157 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 x}{128}-\frac {9 a^3 \cos ^5(c+d x)}{80 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac {9 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d} \] Output:
27/128*a^3*x-9/80*a^3*cos(d*x+c)^5/d+27/128*a^3*cos(d*x+c)*sin(d*x+c)/d+9/ 64*a^3*cos(d*x+c)^3*sin(d*x+c)/d-3/56*a*cos(d*x+c)^5*(a+a*sin(d*x+c))^2/d- 1/8*cos(d*x+c)^5*(a+a*sin(d*x+c))^3/d-9/112*cos(d*x+c)^5*(a^3+a^3*sin(d*x+ c))/d
Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (8400 c+7560 d x-9520 \cos (c+d x)-3920 \cos (3 (c+d x))-112 \cos (5 (c+d x))+240 \cos (7 (c+d x))+1680 \sin (2 (c+d x))-1960 \sin (4 (c+d x))-560 \sin (6 (c+d x))+35 \sin (8 (c+d x)))}{35840 d} \] Input:
Integrate[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(8400*c + 7560*d*x - 9520*Cos[c + d*x] - 3920*Cos[3*(c + d*x)] - 112* Cos[5*(c + d*x)] + 240*Cos[7*(c + d*x)] + 1680*Sin[2*(c + d*x)] - 1960*Sin [4*(c + d*x)] - 560*Sin[6*(c + d*x)] + 35*Sin[8*(c + d*x)]))/(35840*d)
Time = 0.78 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3339, 3042, 3157, 3042, 3157, 3042, 3148, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^4 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle \frac {3}{8} \int \cos ^4(c+d x) (\sin (c+d x) a+a)^3dx-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{8} \int \cos (c+d x)^4 (\sin (c+d x) a+a)^3dx-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)^2dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)^2dx-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \int \cos ^4(c+d x) (\sin (c+d x) a+a)dx-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \int \cos (c+d x)^4 (\sin (c+d x) a+a)dx-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \int \cos ^4(c+d x)dx-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {3}{8} \left (\frac {9}{7} a \left (\frac {7}{6} a \left (a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {a \cos ^5(c+d x)}{5 d}\right )-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{6 d}\right )-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}\right )-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d}\) |
Input:
Int[Cos[c + d*x]^4*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
-1/8*(Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/d + (3*(-1/7*(a*Cos[c + d*x]^ 5*(a + a*Sin[c + d*x])^2)/d + (9*a*(-1/6*(Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/d + (7*a*(-1/5*(a*Cos[c + d*x]^5)/d + a*((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))/7))/8
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 0]
Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.13
\[\frac {a^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{7}-\frac {2 \cos \left (d x +c \right )^{5}}{35}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a^{3} \cos \left (d x +c \right )^{5}}{5}}{d}\]
Input:
int(cos(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
1/d*(a^3*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64 *(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+3*a^3*(-1/7*s in(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)+3*a^3*(-1/6*sin(d*x+c)*cos(d*x +c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)-1/5*a ^3*cos(d*x+c)^5)
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.62 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {1920 \, a^{3} \cos \left (d x + c\right )^{7} - 3584 \, a^{3} \cos \left (d x + c\right )^{5} + 945 \, a^{3} d x + 35 \, {\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 88 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{3} \cos \left (d x + c\right )^{3} + 27 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/4480*(1920*a^3*cos(d*x + c)^7 - 3584*a^3*cos(d*x + c)^5 + 945*a^3*d*x + 35*(16*a^3*cos(d*x + c)^7 - 88*a^3*cos(d*x + c)^5 + 18*a^3*cos(d*x + c)^3 + 27*a^3*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (148) = 296\).
Time = 0.76 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.80 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {11 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {6 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)*(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((3*a**3*x*sin(c + d*x)**8/128 + 3*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 3*a**3*x*sin(c + d*x)**6/16 + 9*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 9*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*a**3*x*sin (c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4 /16 + 3*a**3*x*cos(c + d*x)**8/128 + 3*a**3*x*cos(c + d*x)**6/16 + 3*a**3* sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*a**3*sin(c + d*x)**5*cos(c + d*x )**3/(128*d) + 3*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 11*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) + a**3*sin(c + d*x)**3*cos(c + d*x)**3/( 2*d) - 3*a**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**3*sin(c + d*x)* cos(c + d*x)**7/(128*d) - 3*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 6*a **3*cos(c + d*x)**7/(35*d) - a**3*cos(c + d*x)**5/(5*d), Ne(d, 0)), (x*(a* sin(c) + a)**3*sin(c)*cos(c)**4, True))
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.73 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {7168 \, a^{3} \cos \left (d x + c\right )^{5} - 3072 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 35 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{35840 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/35840*(7168*a^3*cos(d*x + c)^5 - 3072*(5*cos(d*x + c)^7 - 7*cos(d*x + c )^5)*a^3 - 560*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c)) *a^3 - 35*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^3)/d
Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27}{128} \, a^{3} x + \frac {3 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {7 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {17 \, a^{3} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
27/128*a^3*x + 3/448*a^3*cos(7*d*x + 7*c)/d - 1/320*a^3*cos(5*d*x + 5*c)/d - 7/64*a^3*cos(3*d*x + 3*c)/d - 17/64*a^3*cos(d*x + c)/d + 1/1024*a^3*sin (8*d*x + 8*c)/d - 1/64*a^3*sin(6*d*x + 6*c)/d - 7/128*a^3*sin(4*d*x + 4*c) /d + 3/64*a^3*sin(2*d*x + 2*c)/d
Time = 19.18 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.94 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^4*sin(c + d*x)*(a + a*sin(c + d*x))^3,x)
Output:
(27*a^3*x)/128 - ((919*a^3*tan(c/2 + (d*x)/2)^7)/64 - (437*a^3*tan(c/2 + ( d*x)/2)^5)/64 - (305*a^3*tan(c/2 + (d*x)/2)^3)/64 - (919*a^3*tan(c/2 + (d* x)/2)^9)/64 + (437*a^3*tan(c/2 + (d*x)/2)^11)/64 + (305*a^3*tan(c/2 + (d*x )/2)^13)/64 - (27*a^3*tan(c/2 + (d*x)/2)^15)/64 + (a^3*(945*c + 945*d*x))/ 4480 - (a^3*(945*c + 945*d*x - 3328))/4480 + tan(c/2 + (d*x)/2)^14*((a^3*( 945*c + 945*d*x))/560 - (a^3*(7560*c + 7560*d*x - 8960))/4480) + tan(c/2 + (d*x)/2)^2*((a^3*(945*c + 945*d*x))/560 - (a^3*(7560*c + 7560*d*x - 17664 ))/4480) + tan(c/2 + (d*x)/2)^4*((a^3*(945*c + 945*d*x))/160 - (a^3*(26460 *c + 26460*d*x - 12544))/4480) + tan(c/2 + (d*x)/2)^12*((a^3*(945*c + 945* d*x))/160 - (a^3*(26460*c + 26460*d*x - 80640))/4480) + tan(c/2 + (d*x)/2) ^10*((a^3*(945*c + 945*d*x))/80 - (a^3*(52920*c + 52920*d*x - 44800))/4480 ) + tan(c/2 + (d*x)/2)^6*((a^3*(945*c + 945*d*x))/80 - (a^3*(52920*c + 529 20*d*x - 141568))/4480) + tan(c/2 + (d*x)/2)^8*((a^3*(945*c + 945*d*x))/64 - (a^3*(66150*c + 66150*d*x - 116480))/4480) + (27*a^3*tan(c/2 + (d*x)/2) )/64)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^8)
Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-1400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+2176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+3850 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1408 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-945 \cos \left (d x +c \right ) \sin \left (d x +c \right )-1664 \cos \left (d x +c \right )+945 d x +1664\right )}{4480 d} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 560*cos(c + d*x)*sin(c + d*x)**7 - 1920*cos(c + d*x)*sin(c + d*x )**6 - 1400*cos(c + d*x)*sin(c + d*x)**5 + 2176*cos(c + d*x)*sin(c + d*x)* *4 + 3850*cos(c + d*x)*sin(c + d*x)**3 + 1408*cos(c + d*x)*sin(c + d*x)**2 - 945*cos(c + d*x)*sin(c + d*x) - 1664*cos(c + d*x) + 945*d*x + 1664))/(4 480*d)