Integrand size = 27, antiderivative size = 113 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 d} \] Output:
-1/8*a*cos(d*x+c)^8/d+1/5*a*cos(d*x+c)^10/d-1/12*a*cos(d*x+c)^12/d+1/7*a*s in(d*x+c)^7/d-1/3*a*sin(d*x+c)^9/d+3/11*a*sin(d*x+c)^11/d-1/13*a*sin(d*x+c )^13/d
Time = 0.70 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (600600 \cos (2 (c+d x))+75075 \cos (4 (c+d x))-100100 \cos (6 (c+d x))-30030 \cos (8 (c+d x))+12012 \cos (10 (c+d x))+5005 \cos (12 (c+d x))-600600 \sin (c+d x)+150150 \sin (3 (c+d x))+90090 \sin (5 (c+d x))-25740 \sin (7 (c+d x))-20020 \sin (9 (c+d x))+2730 \sin (11 (c+d x))+2310 \sin (13 (c+d x)))}{123002880 d} \] Input:
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]
Output:
-1/123002880*(a*(600600*Cos[2*(c + d*x)] + 75075*Cos[4*(c + d*x)] - 100100 *Cos[6*(c + d*x)] - 30030*Cos[8*(c + d*x)] + 12012*Cos[10*(c + d*x)] + 500 5*Cos[12*(c + d*x)] - 600600*Sin[c + d*x] + 150150*Sin[3*(c + d*x)] + 9009 0*Sin[5*(c + d*x)] - 25740*Sin[7*(c + d*x)] - 20020*Sin[9*(c + d*x)] + 273 0*Sin[11*(c + d*x)] + 2310*Sin[13*(c + d*x)]))/d
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3313, 3042, 3044, 244, 2009, 3045, 243, 49, 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.
| \(\displaystyle \int \sin ^5(c+d x) \cos ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^5 \cos (c+d x)^7 (a \sin (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3313 |
| \(\displaystyle a \int \cos ^7(c+d x) \sin ^6(c+d x)dx+a \int \cos ^7(c+d x) \sin ^5(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^5dx+a \int \cos (c+d x)^7 \sin (c+d x)^6dx\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \int \sin ^6(c+d x) \left (1-\sin ^2(c+d x)\right )^3d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^5dx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {a \int \left (-\sin ^{12}(c+d x)+3 \sin ^{10}(c+d x)-3 \sin ^8(c+d x)+\sin ^6(c+d x)\right )d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^5dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^5dx+\frac {a \left (-\frac {1}{13} \sin ^{13}(c+d x)+\frac {3}{11} \sin ^{11}(c+d x)-\frac {1}{3} \sin ^9(c+d x)+\frac {1}{7} \sin ^7(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {a \left (-\frac {1}{13} \sin ^{13}(c+d x)+\frac {3}{11} \sin ^{11}(c+d x)-\frac {1}{3} \sin ^9(c+d x)+\frac {1}{7} \sin ^7(c+d x)\right )}{d}-\frac {a \int \cos ^7(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {a \left (-\frac {1}{13} \sin ^{13}(c+d x)+\frac {3}{11} \sin ^{11}(c+d x)-\frac {1}{3} \sin ^9(c+d x)+\frac {1}{7} \sin ^7(c+d x)\right )}{d}-\frac {a \int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {a \left (-\frac {1}{13} \sin ^{13}(c+d x)+\frac {3}{11} \sin ^{11}(c+d x)-\frac {1}{3} \sin ^9(c+d x)+\frac {1}{7} \sin ^7(c+d x)\right )}{d}-\frac {a \int \left (\cos ^{10}(c+d x)-2 \cos ^8(c+d x)+\cos ^6(c+d x)\right )d\cos ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \left (-\frac {1}{13} \sin ^{13}(c+d x)+\frac {3}{11} \sin ^{11}(c+d x)-\frac {1}{3} \sin ^9(c+d x)+\frac {1}{7} \sin ^7(c+d x)\right )}{d}-\frac {a \left (\frac {1}{6} \cos ^{12}(c+d x)-\frac {2}{5} \cos ^{10}(c+d x)+\frac {1}{4} \cos ^8(c+d x)\right )}{2 d}\) |
Input:
Int[Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]),x]
Output:
-1/2*(a*(Cos[c + d*x]^8/4 - (2*Cos[c + d*x]^10)/5 + Cos[c + d*x]^12/6))/d + (a*(Sin[c + d*x]^7/7 - Sin[c + d*x]^9/3 + (3*Sin[c + d*x]^11)/11 - Sin[c + d*x]^13/13))/d
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[Cos[e + f*x]^ p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[Cos[e + f*x]^p*(d*Sin[e + f*x ])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2 ] && IntegerQ[n] && ((LtQ[p, 0] && NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] | | LtQ[p + 1, -n, 2*p + 1])
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
\[-\frac {a \left (\frac {\sin \left (d x +c \right )^{13}}{13}+\frac {\sin \left (d x +c \right )^{12}}{12}-\frac {3 \sin \left (d x +c \right )^{11}}{11}-\frac {3 \sin \left (d x +c \right )^{10}}{10}+\frac {\sin \left (d x +c \right )^{9}}{3}+\frac {3 \sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{6}}{6}\right )}{d}\]
Input:
int(cos(d*x+c)^7*sin(d*x+c)^5*(a+a*sin(d*x+c)),x)
Output:
-a/d*(1/13*sin(d*x+c)^13+1/12*sin(d*x+c)^12-3/11*sin(d*x+c)^11-3/10*sin(d* x+c)^10+1/3*sin(d*x+c)^9+3/8*sin(d*x+c)^8-1/7*sin(d*x+c)^7-1/6*sin(d*x+c)^ 6)
Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {10010 \, a \cos \left (d x + c\right )^{12} - 24024 \, a \cos \left (d x + c\right )^{10} + 15015 \, a \cos \left (d x + c\right )^{8} + 40 \, {\left (231 \, a \cos \left (d x + c\right )^{12} - 567 \, a \cos \left (d x + c\right )^{10} + 371 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{120120 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/120120*(10010*a*cos(d*x + c)^12 - 24024*a*cos(d*x + c)^10 + 15015*a*cos (d*x + c)^8 + 40*(231*a*cos(d*x + c)^12 - 567*a*cos(d*x + c)^10 + 371*a*co s(d*x + c)^8 - 5*a*cos(d*x + c)^6 - 6*a*cos(d*x + c)^4 - 8*a*cos(d*x + c)^ 2 - 16*a)*sin(d*x + c))/d
Time = 3.43 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.61 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{13}{\left (c + d x \right )}}{3003 d} + \frac {a \sin ^{12}{\left (c + d x \right )}}{120 d} + \frac {8 a \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{231 d} + \frac {a \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{20 d} + \frac {2 a \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{21 d} + \frac {a \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8 d} + \frac {a \sin ^{7}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{7 d} + \frac {a \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{5}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**5*(a+a*sin(d*x+c)),x)
Output:
Piecewise((16*a*sin(c + d*x)**13/(3003*d) + a*sin(c + d*x)**12/(120*d) + 8 *a*sin(c + d*x)**11*cos(c + d*x)**2/(231*d) + a*sin(c + d*x)**10*cos(c + d *x)**2/(20*d) + 2*a*sin(c + d*x)**9*cos(c + d*x)**4/(21*d) + a*sin(c + d*x )**8*cos(c + d*x)**4/(8*d) + a*sin(c + d*x)**7*cos(c + d*x)**6/(7*d) + a*s in(c + d*x)**6*cos(c + d*x)**6/(6*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)* *5*cos(c)**7, True))
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {9240 \, a \sin \left (d x + c\right )^{13} + 10010 \, a \sin \left (d x + c\right )^{12} - 32760 \, a \sin \left (d x + c\right )^{11} - 36036 \, a \sin \left (d x + c\right )^{10} + 40040 \, a \sin \left (d x + c\right )^{9} + 45045 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7} - 20020 \, a \sin \left (d x + c\right )^{6}}{120120 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/120120*(9240*a*sin(d*x + c)^13 + 10010*a*sin(d*x + c)^12 - 32760*a*sin( d*x + c)^11 - 36036*a*sin(d*x + c)^10 + 40040*a*sin(d*x + c)^9 + 45045*a*s in(d*x + c)^8 - 17160*a*sin(d*x + c)^7 - 20020*a*sin(d*x + c)^6)/d
Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {9240 \, a \sin \left (d x + c\right )^{13} + 10010 \, a \sin \left (d x + c\right )^{12} - 32760 \, a \sin \left (d x + c\right )^{11} - 36036 \, a \sin \left (d x + c\right )^{10} + 40040 \, a \sin \left (d x + c\right )^{9} + 45045 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7} - 20020 \, a \sin \left (d x + c\right )^{6}}{120120 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/120120*(9240*a*sin(d*x + c)^13 + 10010*a*sin(d*x + c)^12 - 32760*a*sin( d*x + c)^11 - 36036*a*sin(d*x + c)^10 + 40040*a*sin(d*x + c)^9 + 45045*a*s in(d*x + c)^8 - 17160*a*sin(d*x + c)^7 - 20020*a*sin(d*x + c)^6)/d
Time = 33.79 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{13}}{13}-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}-\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}}{d} \] Input:
int(cos(c + d*x)^7*sin(c + d*x)^5*(a + a*sin(c + d*x)),x)
Output:
((a*sin(c + d*x)^6)/6 + (a*sin(c + d*x)^7)/7 - (3*a*sin(c + d*x)^8)/8 - (a *sin(c + d*x)^9)/3 + (3*a*sin(c + d*x)^10)/10 + (3*a*sin(c + d*x)^11)/11 - (a*sin(c + d*x)^12)/12 - (a*sin(c + d*x)^13)/13)/d
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) \sin ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right )^{6} a \left (-9240 \sin \left (d x +c \right )^{7}-10010 \sin \left (d x +c \right )^{6}+32760 \sin \left (d x +c \right )^{5}+36036 \sin \left (d x +c \right )^{4}-40040 \sin \left (d x +c \right )^{3}-45045 \sin \left (d x +c \right )^{2}+17160 \sin \left (d x +c \right )+20020\right )}{120120 d} \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^5*(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**6*a*( - 9240*sin(c + d*x)**7 - 10010*sin(c + d*x)**6 + 3276 0*sin(c + d*x)**5 + 36036*sin(c + d*x)**4 - 40040*sin(c + d*x)**3 - 45045* sin(c + d*x)**2 + 17160*sin(c + d*x) + 20020))/(120120*d)