Integrand size = 27, antiderivative size = 113 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d} \]
-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 = 0.32 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x))-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x)))}{9461760 d} \]
-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 = 0.44 (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 |
| \(\displaystyle \frac {a \left (-\frac {1}{11} \sin ^{11}(c+d x)+\frac {1}{3} \sin ^9(c+d x)-\frac {3}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \left (\frac {1}{6} \cos ^{12}(c+d x)-\frac {2}{5} \cos ^{10}(c+d x)+\frac {1}{4} \cos ^8(c+d x)\right )}{2 d}\) |
-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
3.7.58.3.1 Defintions of rubi rules used
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 = 0.72 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
| default | \(-\frac {a \left (\frac {\left (\sin ^{12}\left (d x +c \right )\right )}{12}+\frac {\left (\sin ^{11}\left (d x +c \right )\right )}{11}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}\right )}{d}\) | \(88\) |
| parallelrisch | \(\frac {a \left (\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-5 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (26680 \cos \left (2 d x +2 c \right )+385 \sin \left (7 d x +7 c \right )+2849 \sin \left (5 d x +5 c \right )+840 \cos \left (6 d x +6 c \right )+8085 \sin \left (d x +c \right )+8085 \sin \left (3 d x +3 c \right )+7280 \cos \left (4 d x +4 c \right )+24336\right ) \left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+5 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2365440 d}\) | \(147\) |
| risch | \(\frac {a \sin \left (11 d x +11 c \right )}{11264 d}-\frac {a \cos \left (12 d x +12 c \right )}{24576 d}+\frac {7 a \sin \left (d x +c \right )}{512 d}-\frac {a \cos \left (10 d x +10 c \right )}{10240 d}+\frac {a \sin \left (9 d x +9 c \right )}{3072 d}+\frac {a \cos \left (8 d x +8 c \right )}{4096 d}-\frac {a \sin \left (7 d x +7 c \right )}{7168 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{6144 d}-\frac {11 a \sin \left (5 d x +5 c \right )}{5120 d}-\frac {5 a \cos \left (4 d x +4 c \right )}{8192 d}-\frac {a \sin \left (3 d x +3 c \right )}{512 d}-\frac {5 a \cos \left (2 d x +2 c \right )}{1024 d}\) | \(179\) |
-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 = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \cos \left (d x + c\right )^{12} - 1848 \, a \cos \left (d x + c\right )^{10} + 1155 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \]
-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 = 2.63 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{20 d} - \frac {a \cos ^{12}{\left (c + d x \right )}}{120 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{4}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((16*a*sin(c + d*x)**11/(1155*d) + 8*a*sin(c + d*x)**9*cos(c + d* x)**2/(105*d) + 6*a*sin(c + d*x)**7*cos(c + d*x)**4/(35*d) + a*sin(c + d*x )**5*cos(c + d*x)**6/(5*d) - a*sin(c + d*x)**4*cos(c + d*x)**8/(8*d) - a*s in(c + d*x)**2*cos(c + d*x)**10/(20*d) - a*cos(c + d*x)**12/(120*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**4*cos(c)**7, True))
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \]
-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 = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.58 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac {5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, d} + \frac {a \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac {a \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {11 \, a \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{512 \, d} \]
-1/24576*a*cos(12*d*x + 12*c)/d - 1/10240*a*cos(10*d*x + 10*c)/d + 1/4096* a*cos(8*d*x + 8*c)/d + 5/6144*a*cos(6*d*x + 6*c)/d - 5/8192*a*cos(4*d*x + 4*c)/d - 5/1024*a*cos(2*d*x + 2*c)/d + 1/11264*a*sin(11*d*x + 11*c)/d + 1/ 3072*a*sin(9*d*x + 9*c)/d - 1/7168*a*sin(7*d*x + 7*c)/d - 11/5120*a*sin(5* d*x + 5*c)/d - 1/512*a*sin(3*d*x + 3*c)/d + 7/512*a*sin(d*x + c)/d
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}-\frac {a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]