Integrand size = 27, antiderivative size = 163 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{16} b \left (6 a^2+b^2\right ) x-\frac {a \left (2 a^2+33 b^2\right ) \cos ^3(c+d x)}{120 d}+\frac {b \left (6 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{40 d}-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d} \] Output:
1/16*b*(6*a^2+b^2)*x-1/120*a*(2*a^2+33*b^2)*cos(d*x+c)^3/d+1/16*b*(6*a^2+b ^2)*cos(d*x+c)*sin(d*x+c)/d-1/40*(2*a^2+5*b^2)*cos(d*x+c)^3*(a+b*sin(d*x+c ))/d-1/10*a*cos(d*x+c)^3*(a+b*sin(d*x+c))^2/d-1/6*cos(d*x+c)^3*(a+b*sin(d* x+c))^3/d
Time = 0.78 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-120 a \left (2 a^2+3 b^2\right ) \cos (c+d x)-20 \left (4 a^3+3 a b^2\right ) \cos (3 (c+d x))+b \left (36 a b \cos (5 (c+d x))+5 \left (72 a^2 c+18 b^2 c+72 a^2 d x+12 b^2 d x-3 b^2 \sin (2 (c+d x))-3 \left (6 a^2+b^2\right ) \sin (4 (c+d x))+b^2 \sin (6 (c+d x))\right )\right )}{960 d} \] Input:
Integrate[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]
Output:
(-120*a*(2*a^2 + 3*b^2)*Cos[c + d*x] - 20*(4*a^3 + 3*a*b^2)*Cos[3*(c + d*x )] + b*(36*a*b*Cos[5*(c + d*x)] + 5*(72*a^2*c + 18*b^2*c + 72*a^2*d*x + 12 *b^2*d*x - 3*b^2*Sin[2*(c + d*x)] - 3*(6*a^2 + b^2)*Sin[4*(c + d*x)] + b^2 *Sin[6*(c + d*x)])))/(960*d)
Time = 0.85 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3341, 27, 3042, 3341, 3042, 3341, 3042, 3148, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^2 (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{6} \int 3 \cos ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2dx-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \cos ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2dx-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \cos (c+d x)^2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2dx-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (7 a b+\left (2 a^2+5 b^2\right ) \sin (c+d x)\right )dx-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \int \cos (c+d x)^2 (a+b \sin (c+d x)) \left (7 a b+\left (2 a^2+5 b^2\right ) \sin (c+d x)\right )dx-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (5 b \left (6 a^2+b^2\right )+a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )dx-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos (c+d x)^2 \left (5 b \left (6 a^2+b^2\right )+a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )dx-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (5 b \left (6 a^2+b^2\right ) \int \cos ^2(c+d x)dx-\frac {a \left (2 a^2+33 b^2\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (5 b \left (6 a^2+b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {a \left (2 a^2+33 b^2\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (5 b \left (6 a^2+b^2\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a \left (2 a^2+33 b^2\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (5 b \left (6 a^2+b^2\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {a \left (2 a^2+33 b^2\right ) \cos ^3(c+d x)}{3 d}\right )-\frac {\left (2 a^2+5 b^2\right ) \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}\right )-\frac {a \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}\right )-\frac {\cos ^3(c+d x) (a+b \sin (c+d x))^3}{6 d}\) |
Input:
Int[Cos[c + d*x]^2*Sin[c + d*x]*(a + b*Sin[c + d*x])^3,x]
Output:
-1/6*(Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/d + (-1/5*(a*Cos[c + d*x]^3*( a + b*Sin[c + d*x])^2)/d + (-1/4*((2*a^2 + 5*b^2)*Cos[c + d*x]^3*(a + b*Si n[c + d*x]))/d + (-1/3*(a*(2*a^2 + 33*b^2)*Cos[c + d*x]^3)/d + 5*b*(6*a^2 + b^2)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)/5)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*((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[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Time = 71.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.85
| method | result | size |
| parallelrisch | \(\frac {\left (-80 a^{3}-60 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (-90 a^{2} b -15 b^{3}\right ) \sin \left (4 d x +4 c \right )+36 a \,b^{2} \cos \left (5 d x +5 c \right )-15 b^{3} \sin \left (2 d x +2 c \right )+5 b^{3} \sin \left (6 d x +6 c \right )+\left (-240 a^{3}-360 a \,b^{2}\right ) \cos \left (d x +c \right )+360 a^{2} b d x +60 b^{3} d x -320 a^{3}-384 a \,b^{2}}{960 d}\) | \(139\) |
| derivativedivides | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{3}}{3}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(158\) |
| default | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{3}}{3}+3 a^{2} b \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(158\) |
| risch | \(\frac {3 a^{2} b x}{8}+\frac {b^{3} x}{16}-\frac {a^{3} \cos \left (d x +c \right )}{4 d}-\frac {3 a \,b^{2} \cos \left (d x +c \right )}{8 d}+\frac {b^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a \,b^{2} \cos \left (5 d x +5 c \right )}{80 d}-\frac {3 b \sin \left (4 d x +4 c \right ) a^{2}}{32 d}-\frac {b^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d}-\frac {\cos \left (3 d x +3 c \right ) a \,b^{2}}{16 d}-\frac {b^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(166\) |
| norman | \(\frac {-\frac {10 a^{3}+12 a \,b^{2}}{15 d}+\frac {b \left (6 a^{2}+b^{2}\right ) x}{16}-\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {3 \left (2 a^{3}+4 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {4 \left (5 a^{3}+6 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {2 \left (5 a^{3}+12 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {b \left (6 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {b \left (6 a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {15 b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {5 b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {15 b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {3 b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {b \left (6 a^{2}+b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {b \left (18 a^{2}+19 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {b \left (18 a^{2}+19 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {b \left (90 a^{2}-17 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {b \left (90 a^{2}-17 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(476\) |
| orering | \(\text {Expression too large to display}\) | \(3125\) |
Input:
int(cos(d*x+c)^2*sin(d*x+c)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/960*((-80*a^3-60*a*b^2)*cos(3*d*x+3*c)+(-90*a^2*b-15*b^3)*sin(4*d*x+4*c) +36*a*b^2*cos(5*d*x+5*c)-15*b^3*sin(2*d*x+2*c)+5*b^3*sin(6*d*x+6*c)+(-240* a^3-360*a*b^2)*cos(d*x+c)+360*a^2*b*d*x+60*b^3*d*x-320*a^3-384*a*b^2)/d
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.71 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {144 \, a b^{2} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (6 \, a^{2} b + b^{3}\right )} d x + 5 \, {\left (8 \, b^{3} \cos \left (d x + c\right )^{5} - 2 \, {\left (18 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/240*(144*a*b^2*cos(d*x + c)^5 - 80*(a^3 + 3*a*b^2)*cos(d*x + c)^3 + 15*( 6*a^2*b + b^3)*d*x + 5*(8*b^3*cos(d*x + c)^5 - 2*(18*a^2*b + 7*b^3)*cos(d* x + c)^3 + 3*(6*a^2*b + b^3)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (143) = 286\).
Time = 0.37 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.09 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} - \frac {a^{3} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 a^{2} b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{2} b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {3 a^{2} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {a b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {2 a b^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {b^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {b^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \sin {\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*sin(d*x+c)*(a+b*sin(d*x+c))**3,x)
Output:
Piecewise((-a**3*cos(c + d*x)**3/(3*d) + 3*a**2*b*x*sin(c + d*x)**4/8 + 3* a**2*b*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*a**2*b*x*cos(c + d*x)**4/8 + 3*a**2*b*sin(c + d*x)**3*cos(c + d*x)/(8*d) - 3*a**2*b*sin(c + d*x)*cos( c + d*x)**3/(8*d) - a*b**2*sin(c + d*x)**2*cos(c + d*x)**3/d - 2*a*b**2*co s(c + d*x)**5/(5*d) + b**3*x*sin(c + d*x)**6/16 + 3*b**3*x*sin(c + d*x)**4 *cos(c + d*x)**2/16 + 3*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + b**3*x *cos(c + d*x)**6/16 + b**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) - b**3*sin( c + d*x)**3*cos(c + d*x)**3/(6*d) - b**3*sin(c + d*x)*cos(c + d*x)**5/(16* d), Ne(d, 0)), (x*(a + b*sin(c))**3*sin(c)*cos(c)**2, True))
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {320 \, a^{3} \cos \left (d x + c\right )^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b - 192 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a b^{2} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{3}}{960 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/960*(320*a^3*cos(d*x + c)^3 - 90*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^2*b - 192*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a*b^2 + 5*(4*sin(2*d*x + 2*c) ^3 - 12*d*x - 12*c + 3*sin(4*d*x + 4*c))*b^3)/d
Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.85 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3 \, a b^{2} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {b^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{16} \, {\left (6 \, a^{2} b + b^{3}\right )} x - \frac {{\left (4 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )}{8 \, d} - \frac {{\left (6 \, a^{2} b + b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
3/80*a*b^2*cos(5*d*x + 5*c)/d + 1/192*b^3*sin(6*d*x + 6*c)/d - 1/64*b^3*si n(2*d*x + 2*c)/d + 1/16*(6*a^2*b + b^3)*x - 1/48*(4*a^3 + 3*a*b^2)*cos(3*d *x + 3*c)/d - 1/8*(2*a^3 + 3*a*b^2)*cos(d*x + c)/d - 1/64*(6*a^2*b + b^3)* sin(4*d*x + 4*c)/d
Time = 36.13 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.61 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2+b^2\right )}{8\,\left (\frac {3\,a^2\,b}{4}+\frac {b^3}{8}\right )}\right )\,\left (6\,a^2+b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2\,b}{4}+\frac {b^3}{8}\right )+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {4\,a\,b^2}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^3+\frac {24\,a\,b^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {20\,a^3}{3}+8\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {3\,a^2\,b}{4}+\frac {b^3}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a^2\,b}{2}+\frac {19\,b^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {9\,a^2\,b}{2}+\frac {19\,b^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {15\,a^2\,b}{4}-\frac {17\,b^3}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {15\,a^2\,b}{4}-\frac {17\,b^3}{24}\right )+\frac {2\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {b\,\left (6\,a^2+b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \] Input:
int(cos(c + d*x)^2*sin(c + d*x)*(a + b*sin(c + d*x))^3,x)
Output:
(b*atan((b*tan(c/2 + (d*x)/2)*(6*a^2 + b^2))/(8*((3*a^2*b)/4 + b^3/8)))*(6 *a^2 + b^2))/(8*d) - (tan(c/2 + (d*x)/2)*((3*a^2*b)/4 + b^3/8) + 4*a^3*tan (c/2 + (d*x)/2)^4 + 2*a^3*tan(c/2 + (d*x)/2)^10 + (4*a*b^2)/5 + tan(c/2 + (d*x)/2)^8*(12*a*b^2 + 6*a^3) + tan(c/2 + (d*x)/2)^2*((24*a*b^2)/5 + 2*a^3 ) + tan(c/2 + (d*x)/2)^6*(8*a*b^2 + (20*a^3)/3) - tan(c/2 + (d*x)/2)^11*(( 3*a^2*b)/4 + b^3/8) - tan(c/2 + (d*x)/2)^5*((9*a^2*b)/2 + (19*b^3)/4) + ta n(c/2 + (d*x)/2)^7*((9*a^2*b)/2 + (19*b^3)/4) - tan(c/2 + (d*x)/2)^3*((15* a^2*b)/4 - (17*b^3)/24) + tan(c/2 + (d*x)/2)^9*((15*a^2*b)/4 - (17*b^3)/24 ) + (2*a^3)/3)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*t an(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - (b*(6*a^2 + b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)
Time = 0.18 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.27 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{3}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{2}+180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b -10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3}-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{2}-90 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b -15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{3}-80 \cos \left (d x +c \right ) a^{3}-96 \cos \left (d x +c \right ) a \,b^{2}+80 a^{3}+90 a^{2} b d x +96 a \,b^{2}+15 b^{3} d x}{240 d} \] Input:
int(cos(d*x+c)^2*sin(d*x+c)*(a+b*sin(d*x+c))^3,x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5*b**3 + 144*cos(c + d*x)*sin(c + d*x)**4*a *b**2 + 180*cos(c + d*x)*sin(c + d*x)**3*a**2*b - 10*cos(c + d*x)*sin(c + d*x)**3*b**3 + 80*cos(c + d*x)*sin(c + d*x)**2*a**3 - 48*cos(c + d*x)*sin( c + d*x)**2*a*b**2 - 90*cos(c + d*x)*sin(c + d*x)*a**2*b - 15*cos(c + d*x) *sin(c + d*x)*b**3 - 80*cos(c + d*x)*a**3 - 96*cos(c + d*x)*a*b**2 + 80*a* *3 + 90*a**2*b*d*x + 96*a*b**2 + 15*b**3*d*x)/(240*d)