Integrand size = 27, antiderivative size = 113 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d} \] 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+1/5*a*s in(d*x+c)^5/d-3/7*a*sin(d*x+c)^7/d+1/3*a*sin(d*x+c)^9/d-1/11*a*sin(d*x+c)^ 11/d
Time = 0.54 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x))-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x)))}{9461760 d} \] Input:
Integrate[Cos[c + d*x]^7*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]
Output:
-1/9461760*(a*(46200*Cos[2*(c + d*x)] + 5775*Cos[4*(c + d*x)] - 7700*Cos[6 *(c + d*x)] - 2310*Cos[8*(c + d*x)] + 924*Cos[10*(c + d*x)] + 385*Cos[12*( c + d*x)] - 129360*Sin[c + d*x] + 18480*Sin[3*(c + d*x)] + 20328*Sin[5*(c + d*x)] + 1320*Sin[7*(c + d*x)] - 3080*Sin[9*(c + d*x)] - 840*Sin[11*(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.
\(\displaystyle \int \sin ^4(c+d x) \cos ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^4 \cos (c+d x)^7 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cos ^7(c+d x) \sin ^5(c+d x)dx+a \int \cos ^7(c+d x) \sin ^4(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^4dx+a \int \cos (c+d x)^7 \sin (c+d x)^5dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {a \int \sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )^3d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^5dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \int \left (-\sin ^{10}(c+d x)+3 \sin ^8(c+d x)-3 \sin ^6(c+d x)+\sin ^4(c+d x)\right )d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^5dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^5dx+\frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \cos ^7(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )^2d\cos ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \left (\cos ^{10}(c+d x)-2 \cos ^8(c+d x)+\cos ^6(c+d x)\right )d\cos ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \left (\frac {1}{6} \cos ^{12}(c+d x)-\frac {2}{5} \cos ^{10}(c+d x)+\frac {1}{4} \cos ^8(c+d x)\right )}{2 d}\) |
Input:
Int[Cos[c + d*x]^7*Sin[c + d*x]^4*(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]^5/5 - (3*Sin[c + d*x]^7)/7 + Sin[c + d*x]^9/3 - Sin[c + d*x]^11/11))/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 = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
\[-\frac {a \left (\frac {\sin \left (d x +c \right )^{12}}{12}+\frac {\sin \left (d x +c \right )^{11}}{11}-\frac {3 \sin \left (d x +c \right )^{10}}{10}-\frac {\sin \left (d x +c \right )^{9}}{3}+\frac {3 \sin \left (d x +c \right )^{8}}{8}+\frac {3 \sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{5}}{5}\right )}{d}\]
Input:
int(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x)
Output:
-a/d*(1/12*sin(d*x+c)^12+1/11*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+3/7*sin(d*x+c)^7-1/6*sin(d*x+c)^6-1/5*sin(d*x+c)^5)
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \cos \left (d x + c\right )^{12} - 1848 \, a \cos \left (d x + c\right )^{10} + 1155 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/9240*(770*a*cos(d*x + c)^12 - 1848*a*cos(d*x + c)^10 + 1155*a*cos(d*x + c)^8 - 8*(105*a*cos(d*x + c)^10 - 140*a*cos(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 = 2.53 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.61 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {a \sin ^{12}{\left (c + d x \right )}}{120 d} + \frac {16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {a \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{20 d} + \frac {8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {a \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8 d} + \frac {6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**4*(a+a*sin(d*x+c)),x)
Output:
Piecewise((a*sin(c + d*x)**12/(120*d) + 16*a*sin(c + d*x)**11/(1155*d) + a *sin(c + d*x)**10*cos(c + d*x)**2/(20*d) + 8*a*sin(c + d*x)**9*cos(c + d*x )**2/(105*d) + a*sin(c + d*x)**8*cos(c + d*x)**4/(8*d) + 6*a*sin(c + d*x)* *7*cos(c + d*x)**4/(35*d) + a*sin(c + d*x)**6*cos(c + d*x)**6/(6*d) + a*si n(c + d*x)**5*cos(c + d*x)**6/(5*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)** 4*cos(c)**7, True))
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/9240*(770*a*sin(d*x + c)^12 + 840*a*sin(d*x + c)^11 - 2772*a*sin(d*x + c)^10 - 3080*a*sin(d*x + c)^9 + 3465*a*sin(d*x + c)^8 + 3960*a*sin(d*x + c )^7 - 1540*a*sin(d*x + c)^6 - 1848*a*sin(d*x + c)^5)/d
Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/9240*(770*a*sin(d*x + c)^12 + 840*a*sin(d*x + c)^11 - 2772*a*sin(d*x + c)^10 - 3080*a*sin(d*x + c)^9 + 3465*a*sin(d*x + c)^8 + 3960*a*sin(d*x + c )^7 - 1540*a*sin(d*x + c)^6 - 1848*a*sin(d*x + c)^5)/d
Time = 34.17 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}-\frac {a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \] Input:
int(cos(c + d*x)^7*sin(c + d*x)^4*(a + a*sin(c + d*x)),x)
Output:
((a*sin(c + d*x)^5)/5 + (a*sin(c + d*x)^6)/6 - (3*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 - ( a*sin(c + d*x)^11)/11 - (a*sin(c + d*x)^12)/12)/d
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right )^{5} a \left (-770 \sin \left (d x +c \right )^{7}-840 \sin \left (d x +c \right )^{6}+2772 \sin \left (d x +c \right )^{5}+3080 \sin \left (d x +c \right )^{4}-3465 \sin \left (d x +c \right )^{3}-3960 \sin \left (d x +c \right )^{2}+1540 \sin \left (d x +c \right )+1848\right )}{9240 d} \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^4*(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**5*a*( - 770*sin(c + d*x)**7 - 840*sin(c + d*x)**6 + 2772*si n(c + d*x)**5 + 3080*sin(c + d*x)**4 - 3465*sin(c + d*x)**3 - 3960*sin(c + d*x)**2 + 1540*sin(c + d*x) + 1848))/(9240*d)