Integrand size = 27, antiderivative size = 97 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{10 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{9 d} \]
-1/8*a*cos(d*x+c)^8/d+1/10*a*cos(d*x+c)^10/d+1/3*a*sin(d*x+c)^3/d-3/5*a*si n(d*x+c)^5/d+3/7*a*sin(d*x+c)^7/d-1/9*a*sin(d*x+c)^9/d
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (4410 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-315 \cos (6 (c+d x))-315 \cos (8 (c+d x))-63 \cos (10 (c+d x))-17640 \sin (c+d x)+2016 \sin (5 (c+d x))+900 \sin (7 (c+d x))+140 \sin (9 (c+d x)))}{322560 d} \]
-1/322560*(a*(4410*Cos[2*(c + d*x)] + 1260*Cos[4*(c + d*x)] - 315*Cos[6*(c + d*x)] - 315*Cos[8*(c + d*x)] - 63*Cos[10*(c + d*x)] - 17640*Sin[c + d*x ] + 2016*Sin[5*(c + d*x)] + 900*Sin[7*(c + d*x)] + 140*Sin[9*(c + d*x)]))/ d
Time = 0.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3313, 3042, 3044, 244, 2009, 3045, 244, 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.
\(\displaystyle \int \sin ^2(c+d x) \cos ^7(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^7 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cos ^7(c+d x) \sin ^3(c+d x)dx+a \int \cos ^7(c+d x) \sin ^2(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^2dx+a \int \cos (c+d x)^7 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {a \int \sin ^2(c+d x) \left (1-\sin ^2(c+d x)\right )^3d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \int \left (-\sin ^8(c+d x)+3 \sin ^6(c+d x)-3 \sin ^4(c+d x)+\sin ^2(c+d x)\right )d\sin (c+d x)}{d}+a \int \cos (c+d x)^7 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^7 \sin (c+d x)^3dx+\frac {a \left (-\frac {1}{9} \sin ^9(c+d x)+\frac {3}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\frac {1}{3} \sin ^3(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {a \left (-\frac {1}{9} \sin ^9(c+d x)+\frac {3}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\frac {1}{3} \sin ^3(c+d x)\right )}{d}-\frac {a \int \cos ^7(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \left (-\frac {1}{9} \sin ^9(c+d x)+\frac {3}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\frac {1}{3} \sin ^3(c+d x)\right )}{d}-\frac {a \int \left (\cos ^7(c+d x)-\cos ^9(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (-\frac {1}{9} \sin ^9(c+d x)+\frac {3}{7} \sin ^7(c+d x)-\frac {3}{5} \sin ^5(c+d x)+\frac {1}{3} \sin ^3(c+d x)\right )}{d}-\frac {a \left (\frac {1}{8} \cos ^8(c+d x)-\frac {1}{10} \cos ^{10}(c+d x)\right )}{d}\) |
-((a*(Cos[c + d*x]^8/8 - Cos[c + d*x]^10/10))/d) + (a*(Sin[c + d*x]^3/3 - (3*Sin[c + d*x]^5)/5 + (3*Sin[c + d*x]^7)/7 - Sin[c + d*x]^9/9))/d
3.7.60.3.1 Defintions of rubi rules used
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 = 0.45 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(88\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}\right )}{d}\) | \(88\) |
parallelrisch | \(-\frac {a \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (5556 \cos \left (2 d x +2 c \right )+63 \sin \left (7 d x +7 c \right )+504 \sin \left (5 d x +5 c \right )+140 \cos \left (6 d x +6 c \right )+2205 \sin \left (d x +c \right )+1638 \sin \left (3 d x +3 c \right )+1320 \cos \left (4 d x +4 c \right )+6424\right ) \left (\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80640 d}\) | \(125\) |
risch | \(\frac {7 a \sin \left (d x +c \right )}{128 d}+\frac {a \cos \left (10 d x +10 c \right )}{5120 d}-\frac {a \sin \left (9 d x +9 c \right )}{2304 d}+\frac {a \cos \left (8 d x +8 c \right )}{1024 d}-\frac {5 a \sin \left (7 d x +7 c \right )}{1792 d}+\frac {a \cos \left (6 d x +6 c \right )}{1024 d}-\frac {a \sin \left (5 d x +5 c \right )}{160 d}-\frac {a \cos \left (4 d x +4 c \right )}{256 d}-\frac {7 a \cos \left (2 d x +2 c \right )}{512 d}\) | \(134\) |
-a/d*(1/10*sin(d*x+c)^10+1/9*sin(d*x+c)^9-3/8*sin(d*x+c)^8-3/7*sin(d*x+c)^ 7+1/2*sin(d*x+c)^6+3/5*sin(d*x+c)^5-1/4*sin(d*x+c)^4-1/3*sin(d*x+c)^3)
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {252 \, a \cos \left (d x + c\right )^{10} - 315 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{2520 \, d} \]
1/2520*(252*a*cos(d*x + c)^10 - 315*a*cos(d*x + c)^8 - 8*(35*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 = 1.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \cos ^{10}{\left (c + d x \right )}}{40 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((16*a*sin(c + d*x)**9/(315*d) + 8*a*sin(c + d*x)**7*cos(c + d*x) **2/(35*d) + 2*a*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + a*sin(c + d*x)**3 *cos(c + d*x)**6/(3*d) - a*sin(c + d*x)**2*cos(c + d*x)**8/(8*d) - a*cos(c + d*x)**10/(40*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**2*cos(c)**7, True ))
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {252 \, a \sin \left (d x + c\right )^{10} + 280 \, a \sin \left (d x + c\right )^{9} - 945 \, a \sin \left (d x + c\right )^{8} - 1080 \, a \sin \left (d x + c\right )^{7} + 1260 \, a \sin \left (d x + c\right )^{6} + 1512 \, a \sin \left (d x + c\right )^{5} - 630 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3}}{2520 \, d} \]
-1/2520*(252*a*sin(d*x + c)^10 + 280*a*sin(d*x + c)^9 - 945*a*sin(d*x + c) ^8 - 1080*a*sin(d*x + c)^7 + 1260*a*sin(d*x + c)^6 + 1512*a*sin(d*x + c)^5 - 630*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^3)/d
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \cos \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} - \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {5 \, a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{160 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{128 \, d} \]
1/5120*a*cos(10*d*x + 10*c)/d + 1/1024*a*cos(8*d*x + 8*c)/d + 1/1024*a*cos (6*d*x + 6*c)/d - 1/256*a*cos(4*d*x + 4*c)/d - 7/512*a*cos(2*d*x + 2*c)/d - 1/2304*a*sin(9*d*x + 9*c)/d - 5/1792*a*sin(7*d*x + 7*c)/d - 1/160*a*sin( 5*d*x + 5*c)/d + 7/128*a*sin(d*x + c)/d
Time = 10.65 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \cos ^7(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{10}}{10}-\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \]