Integrand size = 29, antiderivative size = 187 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 a b x}{128}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {b^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d} \] Output:
3/128*a*b*x-1/7*(a^2+b^2)*cos(d*x+c)^7/d+1/9*(a^2+2*b^2)*cos(d*x+c)^9/d-1/ 11*b^2*cos(d*x+c)^11/d+3/128*a*b*cos(d*x+c)*sin(d*x+c)/d+1/64*a*b*cos(d*x+ c)^3*sin(d*x+c)/d+1/80*a*b*cos(d*x+c)^5*sin(d*x+c)/d-3/40*a*b*cos(d*x+c)^7 *sin(d*x+c)/d-1/5*a*b*cos(d*x+c)^7*sin(d*x+c)^3/d
Time = 1.04 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.05 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {83160 a b c+83160 a b d x-6930 \left (12 a^2+5 b^2\right ) \cos (c+d x)-2310 \left (16 a^2+5 b^2\right ) \cos (3 (c+d x))+3465 b^2 \cos (5 (c+d x))+5940 a^2 \cos (7 (c+d x))+2475 b^2 \cos (7 (c+d x))+1540 a^2 \cos (9 (c+d x))-385 b^2 \cos (9 (c+d x))-315 b^2 \cos (11 (c+d x))+13860 a b \sin (2 (c+d x))-27720 a b \sin (4 (c+d x))-6930 a b \sin (6 (c+d x))+3465 a b \sin (8 (c+d x))+1386 a b \sin (10 (c+d x))}{3548160 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]
Output:
(83160*a*b*c + 83160*a*b*d*x - 6930*(12*a^2 + 5*b^2)*Cos[c + d*x] - 2310*( 16*a^2 + 5*b^2)*Cos[3*(c + d*x)] + 3465*b^2*Cos[5*(c + d*x)] + 5940*a^2*Co s[7*(c + d*x)] + 2475*b^2*Cos[7*(c + d*x)] + 1540*a^2*Cos[9*(c + d*x)] - 3 85*b^2*Cos[9*(c + d*x)] - 315*b^2*Cos[11*(c + d*x)] + 13860*a*b*Sin[2*(c + d*x)] - 27720*a*b*Sin[4*(c + d*x)] - 6930*a*b*Sin[6*(c + d*x)] + 3465*a*b *Sin[8*(c + d*x)] + 1386*a*b*Sin[10*(c + d*x)])/(3548160*d)
Time = 1.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {3042, 3390, 3042, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24, 3680, 354, 86, 2009}
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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^6 (a+b \sin (c+d x))^2dx\) |
| \(\Big \downarrow \) 3390 |
| \(\displaystyle \int \cos ^6(c+d x) \sin ^3(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right )dx+2 a b \int \cos ^6(c+d x) \sin ^4(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \int \cos (c+d x)^6 \sin (c+d x)^4dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \int \cos (c+d x)^6 \sin (c+d x)^3 \left (a^2+b^2 \sin (c+d x)^2\right )dx+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 3680 |
| \(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \sin ^3(c+d x) \left (1-\sin ^2(c+d x)\right )^{5/2} \left (a^2+b^2 \sin ^2(c+d x)\right )d\sin (c+d x)}{d}+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \sin ^2(c+d x) \left (1-\sin ^2(c+d x)\right )^{5/2} \left (a^2+b^2 \sin ^2(c+d x)\right )d\sin ^2(c+d x)}{2 d}+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \int \left (b^2 \left (1-\sin ^2(c+d x)\right )^{9/2}+\left (-a^2-2 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{7/2}+\left (a^2+b^2\right ) \left (1-\sin ^2(c+d x)\right )^{5/2}\right )d\sin ^2(c+d x)}{2 d}+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \left (\frac {2}{9} \left (a^2+2 b^2\right ) \left (1-\sin ^2(c+d x)\right )^{9/2}-\frac {2}{7} \left (a^2+b^2\right ) \left (1-\sin ^2(c+d x)\right )^{7/2}-\frac {2}{11} b^2 \left (1-\sin ^2(c+d x)\right )^{11/2}\right )}{2 d}+2 a b \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]
Output:
(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*((-2*(a^2 + b^2)*(1 - Sin[c + d*x]^2)^( 7/2))/7 + (2*(a^2 + 2*b^2)*(1 - Sin[c + d*x]^2)^(9/2))/9 - (2*b^2*(1 - Sin [c + d*x]^2)^(11/2))/11))/(2*d) + 2*a*b*(-1/10*(Cos[c + d*x]^7*Sin[c + d*x ]^3)/d + (3*(-1/8*(Cos[c + d*x]^7*Sin[c + d*x])/d + ((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[ c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/8))/10)
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[2*a*(b/d) Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e + f* x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFac tors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) S ubst[Int[(d*ff*x)^n*(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, S in[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[m/2]
Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.91
\[\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+2 a b \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{2} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{7}}{11}-\frac {4 \cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{99}-\frac {8 \cos \left (d x +c \right )^{7}}{693}\right )}{d}\]
Input:
int(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x)
Output:
1/d*(a^2*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7)+2*a*b*(-1/10*s in(d*x+c)^3*cos(d*x+c)^7-3/80*cos(d*x+c)^7*sin(d*x+c)+1/160*(cos(d*x+c)^5+ 5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+b^2*(-1/11 *sin(d*x+c)^4*cos(d*x+c)^7-4/99*cos(d*x+c)^7*sin(d*x+c)^2-8/693*cos(d*x+c) ^7))
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {40320 \, b^{2} \cos \left (d x + c\right )^{11} - 49280 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 63360 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} - 10395 \, a b d x - 693 \, {\left (128 \, a b \cos \left (d x + c\right )^{9} - 176 \, a b \cos \left (d x + c\right )^{7} + 8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/443520*(40320*b^2*cos(d*x + c)^11 - 49280*(a^2 + 2*b^2)*cos(d*x + c)^9 + 63360*(a^2 + b^2)*cos(d*x + c)^7 - 10395*a*b*d*x - 693*(128*a*b*cos(d*x + c)^9 - 176*a*b*cos(d*x + c)^7 + 8*a*b*cos(d*x + c)^5 + 10*a*b*cos(d*x + c)^3 + 15*a*b*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (178) = 356\).
Time = 1.80 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.05 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {3 a b x \sin ^{10}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a b x \cos ^{10}{\left (c + d x \right )}}{128} + \frac {3 a b \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {7 a b \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a b \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {3 a b \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{128 d} - \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {8 b^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**3*(a+b*sin(d*x+c))**2,x)
Output:
Piecewise((-a**2*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 2*a**2*cos(c + d* x)**9/(63*d) + 3*a*b*x*sin(c + d*x)**10/128 + 15*a*b*x*sin(c + d*x)**8*cos (c + d*x)**2/128 + 15*a*b*x*sin(c + d*x)**6*cos(c + d*x)**4/64 + 15*a*b*x* sin(c + d*x)**4*cos(c + d*x)**6/64 + 15*a*b*x*sin(c + d*x)**2*cos(c + d*x) **8/128 + 3*a*b*x*cos(c + d*x)**10/128 + 3*a*b*sin(c + d*x)**9*cos(c + d*x )/(128*d) + 7*a*b*sin(c + d*x)**7*cos(c + d*x)**3/(64*d) + a*b*sin(c + d*x )**5*cos(c + d*x)**5/(5*d) - 7*a*b*sin(c + d*x)**3*cos(c + d*x)**7/(64*d) - 3*a*b*sin(c + d*x)*cos(c + d*x)**9/(128*d) - b**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 4*b**2*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 8*b**2*c os(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**3*cos(c)* *6, True))
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {56320 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} + 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 5120 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{3548160 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
1/3548160*(56320*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^2 + 693*(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*b - 5120*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7) *b^2)/d
Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3}{128} \, a b x - \frac {b^{2} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {b^{2} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} + \frac {a b \sin \left (10 \, d x + 10 \, c\right )}{2560 \, d} + \frac {a b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{512 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{256 \, d} + \frac {{\left (4 \, a^{2} - b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {{\left (16 \, a^{2} + 5 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )}{512 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
3/128*a*b*x - 1/11264*b^2*cos(11*d*x + 11*c)/d + 1/1024*b^2*cos(5*d*x + 5* c)/d + 1/2560*a*b*sin(10*d*x + 10*c)/d + 1/1024*a*b*sin(8*d*x + 8*c)/d - 1 /512*a*b*sin(6*d*x + 6*c)/d - 1/128*a*b*sin(4*d*x + 4*c)/d + 1/256*a*b*sin (2*d*x + 2*c)/d + 1/9216*(4*a^2 - b^2)*cos(9*d*x + 9*c)/d + 1/7168*(12*a^2 + 5*b^2)*cos(7*d*x + 7*c)/d - 1/1536*(16*a^2 + 5*b^2)*cos(3*d*x + 3*c)/d - 1/512*(12*a^2 + 5*b^2)*cos(d*x + c)/d
Time = 23.15 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.06 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3\,a\,b\,x}{128}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {4\,a^2}{3}+\frac {32\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (8\,a^2-48\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {64\,a^2}{7}-\frac {48\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {32\,a^2}{3}-\frac {80\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {44\,a^2}{63}+\frac {16\,b^2}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {32\,a^2}{63}-\frac {80\,b^2}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {64\,a^2}{3}+\frac {176\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {72\,a^2}{7}+\frac {240\,b^2}{7}\right )+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {4\,a^2}{63}+\frac {16\,b^2}{693}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}+\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)^3*(a + b*sin(c + d*x))^2,x)
Output:
(3*a*b*x)/128 - (tan(c/2 + (d*x)/2)^16*((4*a^2)/3 + (32*b^2)/3) + tan(c/2 + (d*x)/2)^10*(8*a^2 - 48*b^2) + tan(c/2 + (d*x)/2)^6*((64*a^2)/7 - (48*b^ 2)/7) + tan(c/2 + (d*x)/2)^14*((32*a^2)/3 - (80*b^2)/3) + tan(c/2 + (d*x)/ 2)^2*((44*a^2)/63 + (16*b^2)/63) - tan(c/2 + (d*x)/2)^4*((32*a^2)/63 - (80 *b^2)/63) + tan(c/2 + (d*x)/2)^12*((64*a^2)/3 + (176*b^2)/3) + tan(c/2 + ( d*x)/2)^8*((72*a^2)/7 + (240*b^2)/7) + 4*a^2*tan(c/2 + (d*x)/2)^18 + (4*a^ 2)/63 + (16*b^2)/693 + (a*b*tan(c/2 + (d*x)/2)^3)/2 - (3323*a*b*tan(c/2 + (d*x)/2)^5)/320 + (108*a*b*tan(c/2 + (d*x)/2)^7)/5 - (841*a*b*tan(c/2 + (d *x)/2)^9)/32 + (841*a*b*tan(c/2 + (d*x)/2)^13)/32 - (108*a*b*tan(c/2 + (d* x)/2)^15)/5 + (3323*a*b*tan(c/2 + (d*x)/2)^17)/320 - (a*b*tan(c/2 + (d*x)/ 2)^19)/2 - (3*a*b*tan(c/2 + (d*x)/2)^21)/64 + (3*a*b*tan(c/2 + (d*x)/2))/6 4)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^11)
Time = 0.19 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.62 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {40320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{10} b^{2}+88704 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9} a b +49280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a^{2}-103040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{2}-232848 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a b -133760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}+72320 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}+171864 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b +105600 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}-6930 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b -7040 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-2560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-10395 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -14080 \cos \left (d x +c \right ) a^{2}-5120 \cos \left (d x +c \right ) b^{2}+14080 a^{2}+10395 a b d x +5120 b^{2}}{443520 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x)
Output:
(40320*cos(c + d*x)*sin(c + d*x)**10*b**2 + 88704*cos(c + d*x)*sin(c + d*x )**9*a*b + 49280*cos(c + d*x)*sin(c + d*x)**8*a**2 - 103040*cos(c + d*x)*s in(c + d*x)**8*b**2 - 232848*cos(c + d*x)*sin(c + d*x)**7*a*b - 133760*cos (c + d*x)*sin(c + d*x)**6*a**2 + 72320*cos(c + d*x)*sin(c + d*x)**6*b**2 + 171864*cos(c + d*x)*sin(c + d*x)**5*a*b + 105600*cos(c + d*x)*sin(c + d*x )**4*a**2 - 1920*cos(c + d*x)*sin(c + d*x)**4*b**2 - 6930*cos(c + d*x)*sin (c + d*x)**3*a*b - 7040*cos(c + d*x)*sin(c + d*x)**2*a**2 - 2560*cos(c + d *x)*sin(c + d*x)**2*b**2 - 10395*cos(c + d*x)*sin(c + d*x)*a*b - 14080*cos (c + d*x)*a**2 - 5120*cos(c + d*x)*b**2 + 14080*a**2 + 10395*a*b*d*x + 512 0*b**2)/(443520*d)