Integrand size = 29, antiderivative size = 232 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{16} a \left (2 a^2+3 b^2\right ) x-\frac {b \left (21 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac {b \left (21 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac {a \left (2 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (2 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{56 d}+\frac {b \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{14 d}+\frac {\cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{7 d} \] Output:
1/16*a*(2*a^2+3*b^2)*x-1/35*b*(21*a^2+4*b^2)*cos(d*x+c)/d+1/105*b*(21*a^2+ 4*b^2)*cos(d*x+c)^3/d-1/16*a*(2*a^2+3*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/56*a* (2*a^2-7*b^2)*cos(d*x+c)*sin(d*x+c)^3/d+1/35*b*(a^2-b^2)*cos(d*x+c)*sin(d* x+c)^4/d+1/14*a*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^2/d+1/7*cos(d*x+c )*sin(d*x+c)^3*(a+b*sin(d*x+c))^3/d
Time = 1.83 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.68 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-105 b \left (24 a^2+5 b^2\right ) \cos (c+d x)-35 \left (12 a^2 b+b^3\right ) \cos (3 (c+d x))+63 \left (4 a^2 b+b^3\right ) \cos (5 (c+d x))-15 b^3 \cos (7 (c+d x))+105 a \left (8 a^2 c+12 b^2 c+8 a^2 d x+12 b^2 d x-3 b^2 \sin (2 (c+d x))-\left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+b^2 \sin (6 (c+d x))\right )}{6720 d} \] Input:
Integrate[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]
Output:
(-105*b*(24*a^2 + 5*b^2)*Cos[c + d*x] - 35*(12*a^2*b + b^3)*Cos[3*(c + d*x )] + 63*(4*a^2*b + b^3)*Cos[5*(c + d*x)] - 15*b^3*Cos[7*(c + d*x)] + 105*a *(8*a^2*c + 12*b^2*c + 8*a^2*d*x + 12*b^2*d*x - 3*b^2*Sin[2*(c + d*x)] - ( 2*a^2 + 3*b^2)*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)]))/(6720*d)
Time = 1.42 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3368, 3042, 3529, 3042, 3528, 27, 3042, 3512, 3042, 3502, 3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^2 (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \sin ^2(c+d x) \left (1-\sin ^2(c+d x)\right ) (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \left (1-\sin (c+d x)^2\right ) (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {1}{7} \int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (-3 a \sin ^2(c+d x)+b \sin (c+d x)+4 a\right )dx+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sin (c+d x)^2 (a+b \sin (c+d x))^2 \left (-3 a \sin (c+d x)^2+b \sin (c+d x)+4 a\right )dx+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int 3 \sin ^2(c+d x) (a+b \sin (c+d x)) \left (5 a^2+5 b \sin (c+d x) a-2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )dx+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (5 a^2+5 b \sin (c+d x) a-2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )dx+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \int \sin (c+d x)^2 (a+b \sin (c+d x)) \left (5 a^2+5 b \sin (c+d x) a-2 \left (a^2-b^2\right ) \sin (c+d x)^2\right )dx+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \int \sin ^2(c+d x) \left (25 a^3-5 \left (2 a^2-7 b^2\right ) \sin ^2(c+d x) a+2 b \left (21 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \int \sin (c+d x)^2 \left (25 a^3-5 \left (2 a^2-7 b^2\right ) \sin (c+d x)^2 a+2 b \left (21 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \sin ^2(c+d x) \left (35 a \left (2 a^2+3 b^2\right )+8 b \left (21 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \int \sin (c+d x)^2 \left (35 a \left (2 a^2+3 b^2\right )+8 b \left (21 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (8 b \left (21 a^2+4 b^2\right ) \int \sin ^3(c+d x)dx+35 a \left (2 a^2+3 b^2\right ) \int \sin ^2(c+d x)dx\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (35 a \left (2 a^2+3 b^2\right ) \int \sin (c+d x)^2dx+8 b \left (21 a^2+4 b^2\right ) \int \sin (c+d x)^3dx\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (35 a \left (2 a^2+3 b^2\right ) \int \sin (c+d x)^2dx-\frac {8 b \left (21 a^2+4 b^2\right ) \int \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (35 a \left (2 a^2+3 b^2\right ) \int \sin (c+d x)^2dx-\frac {8 b \left (21 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (35 a \left (2 a^2+3 b^2\right ) \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 b \left (21 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{4} \left (35 a \left (2 a^2+3 b^2\right ) \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 b \left (21 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )+\frac {5 a \left (2 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{2 d}\right )+\frac {\sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}\) |
Input:
Int[Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]
Output:
(Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(7*d) + ((a*Cos[c + d *x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(2*d) + ((2*b*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(5*d) + ((5*a*(2*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d) + ((-8*b*(21*a^2 + 4*b^2)*(Cos[c + d*x] - Cos[c + d*x]^3/3) )/d + 35*a*(2*a^2 + 3*b^2)*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)/5 )/2)/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 246.25 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(\frac {\left (-420 a^{2} b -35 b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (252 a^{2} b +63 b^{3}\right ) \cos \left (5 d x +5 c \right )+\left (-210 a^{3}-315 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )-15 b^{3} \cos \left (7 d x +7 c \right )-315 a \,b^{2} \sin \left (2 d x +2 c \right )+105 a \,b^{2} \sin \left (6 d x +6 c \right )+\left (-2520 a^{2} b -525 b^{3}\right ) \cos \left (d x +c \right )+840 a^{3} d x +1260 a \,b^{2} d x -2688 a^{2} b -512 b^{3}}{6720 d}\) | \(162\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{4}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )}{d}\) | \(196\) |
| default | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}{6}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{4}}{7}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{35}-\frac {8 \cos \left (d x +c \right )^{3}}{105}\right )}{d}\) | \(196\) |
| risch | \(\frac {a^{3} x}{8}+\frac {3 a \,b^{2} x}{16}-\frac {3 a^{2} b \cos \left (d x +c \right )}{8 d}-\frac {5 b^{3} \cos \left (d x +c \right )}{64 d}-\frac {b^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a \,b^{2} \sin \left (6 d x +6 c \right )}{64 d}+\frac {3 b \cos \left (5 d x +5 c \right ) a^{2}}{80 d}+\frac {3 b^{3} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{3} \sin \left (4 d x +4 c \right )}{32 d}-\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{2}}{64 d}-\frac {b \cos \left (3 d x +3 c \right ) a^{2}}{16 d}-\frac {b^{3} \cos \left (3 d x +3 c \right )}{192 d}-\frac {3 a \,b^{2} \sin \left (2 d x +2 c \right )}{64 d}\) | \(202\) |
| norman | \(\frac {-\frac {84 a^{2} b +16 b^{3}}{105 d}+\frac {a \left (2 a^{2}+3 b^{2}\right ) x}{16}-\frac {12 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {\left (24 a^{2} b -16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {\left (24 a^{2} b +16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}-\frac {\left (60 a^{2} b +32 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}-\frac {\left (84 a^{2} b +16 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 d}+\frac {a \left (2 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 d}-\frac {a \left (2 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{2 d}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a \left (2 a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}+\frac {7 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {21 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {35 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}+\frac {35 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {21 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}+\frac {7 a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {a \left (2 a^{2}+3 b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}+\frac {a \left (22 a^{2}+97 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 d}-\frac {a \left (22 a^{2}+97 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(530\) |
| orering | \(\text {Expression too large to display}\) | \(4512\) |
Input:
int(cos(d*x+c)^2*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/6720*((-420*a^2*b-35*b^3)*cos(3*d*x+3*c)+(252*a^2*b+63*b^3)*cos(5*d*x+5* c)+(-210*a^3-315*a*b^2)*sin(4*d*x+4*c)-15*b^3*cos(7*d*x+7*c)-315*a*b^2*sin (2*d*x+2*c)+105*a*b^2*sin(6*d*x+6*c)+(-2520*a^2*b-525*b^3)*cos(d*x+c)+840* a^3*d*x+1260*a*b^2*d*x-2688*a^2*b-512*b^3)/d
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.61 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {240 \, b^{3} \cos \left (d x + c\right )^{7} - 336 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 560 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d x - 105 \, {\left (8 \, a b^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (2 \, a^{3} + 7 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/1680*(240*b^3*cos(d*x + c)^7 - 336*(3*a^2*b + 2*b^3)*cos(d*x + c)^5 + 5 60*(3*a^2*b + b^3)*cos(d*x + c)^3 - 105*(2*a^3 + 3*a*b^2)*d*x - 105*(8*a*b ^2*cos(d*x + c)^5 - 2*(2*a^3 + 7*a*b^2)*cos(d*x + c)^3 + (2*a^3 + 3*a*b^2) *cos(d*x + c))*sin(d*x + c))/d
Time = 0.51 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.70 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {2 a^{2} b \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {8 b^{3} \cos ^{7}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*sin(d*x+c)**2*(a+b*sin(d*x+c))**3,x)
Output:
Piecewise((a**3*x*sin(c + d*x)**4/8 + a**3*x*sin(c + d*x)**2*cos(c + d*x)* *2/4 + a**3*x*cos(c + d*x)**4/8 + a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) - a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) - a**2*b*sin(c + d*x)**2*cos(c + d*x)**3/d - 2*a**2*b*cos(c + d*x)**5/(5*d) + 3*a*b**2*x*sin(c + d*x)**6/1 6 + 9*a*b**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*a*b**2*x*sin(c + d*x )**2*cos(c + d*x)**4/16 + 3*a*b**2*x*cos(c + d*x)**6/16 + 3*a*b**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) - a*b**2*sin(c + d*x)**3*cos(c + d*x)**3/(2*d ) - 3*a*b**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) - b**3*sin(c + d*x)**4*co s(c + d*x)**3/(3*d) - 4*b**3*sin(c + d*x)**2*cos(c + d*x)**5/(15*d) - 8*b* *3*cos(c + d*x)**7/(105*d), Ne(d, 0)), (x*(a + b*sin(c))**3*sin(c)**2*cos( c)**2, True))
Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.56 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {210 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 1344 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 64 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} b^{3}}{6720 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/6720*(210*(4*d*x + 4*c - sin(4*d*x + 4*c))*a^3 + 1344*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^2*b - 105*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3* sin(4*d*x + 4*c))*a*b^2 - 64*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*c os(d*x + c)^3)*b^3)/d
Time = 0.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {b^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a b^{2} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {3 \, a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{16} \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x + \frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (24 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^2*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
-1/448*b^3*cos(7*d*x + 7*c)/d + 1/64*a*b^2*sin(6*d*x + 6*c)/d - 3/64*a*b^2 *sin(2*d*x + 2*c)/d + 1/16*(2*a^3 + 3*a*b^2)*x + 3/320*(4*a^2*b + b^3)*cos (5*d*x + 5*c)/d - 1/192*(12*a^2*b + b^3)*cos(3*d*x + 3*c)/d - 1/64*(24*a^2 *b + 5*b^3)*cos(d*x + c)/d - 1/64*(2*a^3 + 3*a*b^2)*sin(4*d*x + 4*c)/d
Time = 36.30 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.96 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+3\,b^2\right )}{8\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )}\right )\,\left (2\,a^2+3\,b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )+\frac {4\,a^2\,b}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a\,b^2}{2}-a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {5\,a\,b^2}{2}-a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {a^3}{4}+\frac {3\,a\,b^2}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {11\,a^3}{4}+\frac {97\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {11\,a^3}{4}+\frac {97\,a\,b^2}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (8\,a^2\,b-\frac {16\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {24\,a^2\,b}{5}+\frac {16\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (20\,a^2\,b+\frac {32\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {28\,a^2\,b}{5}+\frac {16\,b^3}{15}\right )+\frac {16\,b^3}{105}+12\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (2\,a^2+3\,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)^2*(a + b*sin(c + d*x))^3,x)
Output:
(a*atan((a*tan(c/2 + (d*x)/2)*(2*a^2 + 3*b^2))/(8*((3*a*b^2)/8 + a^3/4)))* (2*a^2 + 3*b^2))/(8*d) - (tan(c/2 + (d*x)/2)*((3*a*b^2)/8 + a^3/4) + (4*a^ 2*b)/5 + tan(c/2 + (d*x)/2)^3*((5*a*b^2)/2 - a^3) - tan(c/2 + (d*x)/2)^11* ((5*a*b^2)/2 - a^3) - tan(c/2 + (d*x)/2)^13*((3*a*b^2)/8 + a^3/4) - tan(c/ 2 + (d*x)/2)^5*((97*a*b^2)/8 + (11*a^3)/4) + tan(c/2 + (d*x)/2)^9*((97*a*b ^2)/8 + (11*a^3)/4) + tan(c/2 + (d*x)/2)^6*(8*a^2*b - (16*b^3)/3) + tan(c/ 2 + (d*x)/2)^4*((24*a^2*b)/5 + (16*b^3)/5) + tan(c/2 + (d*x)/2)^8*(20*a^2* b + (32*b^3)/3) + tan(c/2 + (d*x)/2)^2*((28*a^2*b)/5 + (16*b^3)/15) + (16* b^3)/105 + 12*a^2*b*tan(c/2 + (d*x)/2)^10)/(d*(7*tan(c/2 + (d*x)/2)^2 + 21 *tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^ 14 + 1)) - (a*(2*a^2 + 3*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)
Time = 0.17 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{3}+840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a \,b^{2}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2} b -48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{3}+420 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3}-210 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{2}-336 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b -64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{3}-210 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{2}-672 \cos \left (d x +c \right ) a^{2} b -128 \cos \left (d x +c \right ) b^{3}+210 a^{3} d x +672 a^{2} b +315 a \,b^{2} d x +128 b^{3}}{1680 d} \] Input:
int(cos(d*x+c)^2*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x)
Output:
(240*cos(c + d*x)*sin(c + d*x)**6*b**3 + 840*cos(c + d*x)*sin(c + d*x)**5* a*b**2 + 1008*cos(c + d*x)*sin(c + d*x)**4*a**2*b - 48*cos(c + d*x)*sin(c + d*x)**4*b**3 + 420*cos(c + d*x)*sin(c + d*x)**3*a**3 - 210*cos(c + d*x)* sin(c + d*x)**3*a*b**2 - 336*cos(c + d*x)*sin(c + d*x)**2*a**2*b - 64*cos( c + d*x)*sin(c + d*x)**2*b**3 - 210*cos(c + d*x)*sin(c + d*x)*a**3 - 315*c os(c + d*x)*sin(c + d*x)*a*b**2 - 672*cos(c + d*x)*a**2*b - 128*cos(c + d* x)*b**3 + 210*a**3*d*x + 672*a**2*b + 315*a*b**2*d*x + 128*b**3)/(1680*d)