Integrand size = 29, antiderivative size = 188 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a A \sin (c+d x)}{d}+\frac {(A b+a B) \sin ^2(c+d x)}{2 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}-\frac {3 (A b+a B) \sin ^4(c+d x)}{4 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}+\frac {(A b+a B) \sin ^6(c+d x)}{2 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}-\frac {(A b+a B) \sin ^8(c+d x)}{8 d}-\frac {b B \sin ^9(c+d x)}{9 d} \]
a*A*sin(d*x+c)/d+1/2*(A*b+B*a)*sin(d*x+c)^2/d-1/3*(3*A*a-B*b)*sin(d*x+c)^3 /d-3/4*(A*b+B*a)*sin(d*x+c)^4/d+3/5*(A*a-B*b)*sin(d*x+c)^5/d+1/2*(A*b+B*a) *sin(d*x+c)^6/d-1/7*(A*a-3*B*b)*sin(d*x+c)^7/d-1/8*(A*b+B*a)*sin(d*x+c)^8/ d-1/9*b*B*sin(d*x+c)^9/d
Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {\sin (c+d x) \left (2520 a A+1260 (A b+a B) \sin (c+d x)-840 (3 a A-b B) \sin ^2(c+d x)-1890 (A b+a B) \sin ^3(c+d x)+1512 (a A-b B) \sin ^4(c+d x)+1260 (A b+a B) \sin ^5(c+d x)-360 (a A-3 b B) \sin ^6(c+d x)-315 (A b+a B) \sin ^7(c+d x)-280 b B \sin ^8(c+d x)\right )}{2520 d} \]
(Sin[c + d*x]*(2520*a*A + 1260*(A*b + a*B)*Sin[c + d*x] - 840*(3*a*A - b*B )*Sin[c + d*x]^2 - 1890*(A*b + a*B)*Sin[c + d*x]^3 + 1512*(a*A - b*B)*Sin[ c + d*x]^4 + 1260*(A*b + a*B)*Sin[c + d*x]^5 - 360*(a*A - 3*b*B)*Sin[c + d *x]^6 - 315*(A*b + a*B)*Sin[c + d*x]^7 - 280*b*B*Sin[c + d*x]^8))/(2520*d)
Time = 0.43 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3316, 27, 652, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^7 (a+b \sin (c+d x)) (A+B \sin (c+d x))dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x)) (A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{b}d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \sin (c+d x)) (A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^3d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \frac {\int \left (-b^8 B \sin ^8(c+d x)-b^7 (A b+a B) \sin ^7(c+d x)+b^7 (3 b B-a A) \sin ^6(c+d x)+3 b^7 (A b+a B) \sin ^5(c+d x)-3 b^7 (b B-a A) \sin ^4(c+d x)-3 b^7 (A b+a B) \sin ^3(c+d x)+b^7 (b B-3 a A) \sin ^2(c+d x)+b^7 (A b+a B) \sin (c+d x)+a A b^7\right )d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{8} b^8 (a B+A b) \sin ^8(c+d x)-\frac {1}{7} b^8 (a A-3 b B) \sin ^7(c+d x)+\frac {1}{2} b^8 (a B+A b) \sin ^6(c+d x)+\frac {3}{5} b^8 (a A-b B) \sin ^5(c+d x)-\frac {3}{4} b^8 (a B+A b) \sin ^4(c+d x)-\frac {1}{3} b^8 (3 a A-b B) \sin ^3(c+d x)+\frac {1}{2} b^8 (a B+A b) \sin ^2(c+d x)+a A b^8 \sin (c+d x)-\frac {1}{9} b^9 B \sin ^9(c+d x)}{b^8 d}\) |
(a*A*b^8*Sin[c + d*x] + (b^8*(A*b + a*B)*Sin[c + d*x]^2)/2 - (b^8*(3*a*A - b*B)*Sin[c + d*x]^3)/3 - (3*b^8*(A*b + a*B)*Sin[c + d*x]^4)/4 + (3*b^8*(a *A - b*B)*Sin[c + d*x]^5)/5 + (b^8*(A*b + a*B)*Sin[c + d*x]^6)/2 - (b^8*(a *A - 3*b*B)*Sin[c + d*x]^7)/7 - (b^8*(A*b + a*B)*Sin[c + d*x]^8)/8 - (b^9* B*Sin[c + d*x]^9)/9)/(b^8*d)
3.16.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 1.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(-\frac {\frac {B b \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A b +B a \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (a A -3 B b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 A b -3 B a \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-3 a A +3 B b \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 A b +3 B a \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 a A -B b \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A b -B a \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right ) a}{d}\) | \(159\) |
| default | \(-\frac {\frac {B b \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A b +B a \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (a A -3 B b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 A b -3 B a \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-3 a A +3 B b \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 A b +3 B a \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 a A -B b \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A b -B a \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right ) a}{d}\) | \(159\) |
| parallelrisch | \(\frac {35280 a A \sin \left (3 d x +3 c \right )-2520 A \cos \left (6 d x +6 c \right ) b -8820 A \cos \left (4 d x +4 c \right ) b -17640 A \cos \left (2 d x +2 c \right ) b -315 A \cos \left (8 d x +8 c \right ) b +720 A \sin \left (7 d x +7 c \right ) a +7056 A \sin \left (5 d x +5 c \right ) a +176400 A \sin \left (d x +c \right ) a -2520 B \cos \left (6 d x +6 c \right ) a -8820 B \cos \left (4 d x +4 c \right ) a -17640 B \cos \left (2 d x +2 c \right ) a -140 B b \sin \left (9 d x +9 c \right )-315 B \cos \left (8 d x +8 c \right ) a -900 B \sin \left (7 d x +7 c \right ) b -2016 B \sin \left (5 d x +5 c \right ) b +17640 B b \sin \left (d x +c \right )+29295 A b +29295 B a}{322560 d}\) | \(217\) |
| risch | \(\frac {35 a A \sin \left (d x +c \right )}{64 d}+\frac {7 b B \sin \left (d x +c \right )}{128 d}-\frac {B b \sin \left (9 d x +9 c \right )}{2304 d}-\frac {\cos \left (8 d x +8 c \right ) A b}{1024 d}-\frac {\cos \left (8 d x +8 c \right ) B a}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) a A}{448 d}-\frac {5 \sin \left (7 d x +7 c \right ) B b}{1792 d}-\frac {\cos \left (6 d x +6 c \right ) A b}{128 d}-\frac {\cos \left (6 d x +6 c \right ) B a}{128 d}+\frac {7 \sin \left (5 d x +5 c \right ) a A}{320 d}-\frac {\sin \left (5 d x +5 c \right ) B b}{160 d}-\frac {7 \cos \left (4 d x +4 c \right ) A b}{256 d}-\frac {7 \cos \left (4 d x +4 c \right ) B a}{256 d}+\frac {7 a A \sin \left (3 d x +3 c \right )}{64 d}-\frac {7 \cos \left (2 d x +2 c \right ) A b}{128 d}-\frac {7 \cos \left (2 d x +2 c \right ) B a}{128 d}\) | \(252\) |
| norman | \(\frac {\frac {\left (2 A b +2 B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 A b +2 B a \right ) \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 \left (3 a A +B b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (3 a A +B b \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (17 a A -2 B b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (17 a A -2 B b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (221 a A +79 B b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {8 \left (221 a A +79 B b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 \left (4617 a A -712 B b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (A b +B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (A b +B a \right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a A \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(411\) |
-1/d*(1/9*B*b*sin(d*x+c)^9+1/8*(A*b+B*a)*sin(d*x+c)^8+1/7*(A*a-3*B*b)*sin( d*x+c)^7+1/6*(-3*A*b-3*B*a)*sin(d*x+c)^6+1/5*(-3*A*a+3*B*b)*sin(d*x+c)^5+1 /4*(3*A*b+3*B*a)*sin(d*x+c)^4+1/3*(3*A*a-B*b)*sin(d*x+c)^3+1/2*(-A*b-B*a)* sin(d*x+c)^2-A*sin(d*x+c)*a)
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.56 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {315 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, B b \cos \left (d x + c\right )^{8} - 5 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{6} - 6 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{2} - 144 \, A a - 16 \, B b\right )} \sin \left (d x + c\right )}{2520 \, d} \]
-1/2520*(315*(B*a + A*b)*cos(d*x + c)^8 + 8*(35*B*b*cos(d*x + c)^8 - 5*(9* A*a + B*b)*cos(d*x + c)^6 - 6*(9*A*a + B*b)*cos(d*x + c)^4 - 8*(9*A*a + B* b)*cos(d*x + c)^2 - 144*A*a - 16*B*b)*sin(d*x + c))/d
Time = 0.96 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A b \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {B a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B b \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 B b \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 B b \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B b \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a + b \sin {\left (c \right )}\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((16*A*a*sin(c + d*x)**7/(35*d) + 8*A*a*sin(c + d*x)**5*cos(c + d *x)**2/(5*d) + 2*A*a*sin(c + d*x)**3*cos(c + d*x)**4/d + A*a*sin(c + d*x)* cos(c + d*x)**6/d - A*b*cos(c + d*x)**8/(8*d) - B*a*cos(c + d*x)**8/(8*d) + 16*B*b*sin(c + d*x)**9/(315*d) + 8*B*b*sin(c + d*x)**7*cos(c + d*x)**2/( 35*d) + 2*B*b*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + B*b*sin(c + d*x)**3* cos(c + d*x)**6/(3*d), Ne(d, 0)), (x*(A + B*sin(c))*(a + b*sin(c))*cos(c)* *7, True))
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {280 \, B b \sin \left (d x + c\right )^{9} + 315 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{8} + 360 \, {\left (A a - 3 \, B b\right )} \sin \left (d x + c\right )^{7} - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{6} - 1512 \, {\left (A a - B b\right )} \sin \left (d x + c\right )^{5} + 1890 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{4} + 840 \, {\left (3 \, A a - B b\right )} \sin \left (d x + c\right )^{3} - 2520 \, A a \sin \left (d x + c\right ) - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{2}}{2520 \, d} \]
-1/2520*(280*B*b*sin(d*x + c)^9 + 315*(B*a + A*b)*sin(d*x + c)^8 + 360*(A* a - 3*B*b)*sin(d*x + c)^7 - 1260*(B*a + A*b)*sin(d*x + c)^6 - 1512*(A*a - B*b)*sin(d*x + c)^5 + 1890*(B*a + A*b)*sin(d*x + c)^4 + 840*(3*A*a - B*b)* sin(d*x + c)^3 - 2520*A*a*sin(d*x + c) - 1260*(B*a + A*b)*sin(d*x + c)^2)/ d
Time = 0.40 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {B b \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (B a + A b\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (B a + A b\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, A a - 5 \, B b\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A a - 2 \, B b\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (10 \, A a + B b\right )} \sin \left (d x + c\right )}{128 \, d} \]
-1/2304*B*b*sin(9*d*x + 9*c)/d + 7/64*A*a*sin(3*d*x + 3*c)/d - 1/1024*(B*a + A*b)*cos(8*d*x + 8*c)/d - 1/128*(B*a + A*b)*cos(6*d*x + 6*c)/d - 7/256* (B*a + A*b)*cos(4*d*x + 4*c)/d - 7/128*(B*a + A*b)*cos(2*d*x + 2*c)/d + 1/ 1792*(4*A*a - 5*B*b)*sin(7*d*x + 7*c)/d + 1/320*(7*A*a - 2*B*b)*sin(5*d*x + 5*c)/d + 7/128*(10*A*a + B*b)*sin(d*x + c)/d
Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {B\,b\,{\sin \left (c+d\,x\right )}^9}{9}+\left (\frac {A\,b}{8}+\frac {B\,a}{8}\right )\,{\sin \left (c+d\,x\right )}^8+\left (\frac {A\,a}{7}-\frac {3\,B\,b}{7}\right )\,{\sin \left (c+d\,x\right )}^7+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^6+\left (\frac {3\,B\,b}{5}-\frac {3\,A\,a}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (\frac {3\,A\,b}{4}+\frac {3\,B\,a}{4}\right )\,{\sin \left (c+d\,x\right )}^4+\left (A\,a-\frac {B\,b}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^2-A\,a\,\sin \left (c+d\,x\right )}{d} \]
-(sin(c + d*x)^3*(A*a - (B*b)/3) - sin(c + d*x)^2*((A*b)/2 + (B*a)/2) - si n(c + d*x)^6*((A*b)/2 + (B*a)/2) + sin(c + d*x)^4*((3*A*b)/4 + (3*B*a)/4) - sin(c + d*x)^5*((3*A*a)/5 - (3*B*b)/5) + sin(c + d*x)^7*((A*a)/7 - (3*B* b)/7) + sin(c + d*x)^8*((A*b)/8 + (B*a)/8) - A*a*sin(c + d*x) + (B*b*sin(c + d*x)^9)/9)/d