Integrand size = 27, antiderivative size = 81 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{9 d} \]
-1/6*a*cos(d*x+c)^6/d+1/8*a*cos(d*x+c)^8/d+1/5*a*sin(d*x+c)^5/d-2/7*a*sin( d*x+c)^7/d+1/9*a*sin(d*x+c)^9/d
Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (-7560 \cos (2 (c+d x))-1260 \cos (4 (c+d x))+840 \cos (6 (c+d x))+315 \cos (8 (c+d x))+7560 \sin (c+d x)-1680 \sin (3 (c+d x))-1008 \sin (5 (c+d x))+180 \sin (7 (c+d x))+140 \sin (9 (c+d x)))}{322560 d} \]
(a*(-7560*Cos[2*(c + d*x)] - 1260*Cos[4*(c + d*x)] + 840*Cos[6*(c + d*x)] + 315*Cos[8*(c + d*x)] + 7560*Sin[c + d*x] - 1680*Sin[3*(c + d*x)] - 1008* Sin[5*(c + d*x)] + 180*Sin[7*(c + d*x)] + 140*Sin[9*(c + d*x)]))/(322560*d )
Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91, 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.
\(\displaystyle \int \sin ^3(c+d x) \cos ^5(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^5 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cos ^5(c+d x) \sin ^4(c+d x)dx+a \int \cos ^5(c+d x) \sin ^3(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+a \int \cos (c+d x)^5 \sin (c+d x)^4dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {a \int \sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )^2d\sin (c+d x)}{d}+a \int \cos (c+d x)^5 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \int \left (\sin ^8(c+d x)-2 \sin ^6(c+d x)+\sin ^4(c+d x)\right )d\sin (c+d x)}{d}+a \int \cos (c+d x)^5 \sin (c+d x)^3dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+\frac {a \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {a \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \cos ^5(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {a \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \left (\cos ^5(c+d x)-\cos ^7(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \left (\frac {1}{6} \cos ^6(c+d x)-\frac {1}{8} \cos ^8(c+d x)\right )}{d}\) |
-((a*(Cos[c + d*x]^6/6 - Cos[c + d*x]^8/8))/d) + (a*(Sin[c + d*x]^5/5 - (2 *Sin[c + d*x]^7)/7 + Sin[c + d*x]^9/9))/d
3.5.96.3.1 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 = 0.38 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(67\) |
default | \(\frac {a \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}\right )}{d}\) | \(67\) |
parallelrisch | \(-\frac {a \left (-7665+7560 \cos \left (2 d x +2 c \right )-140 \sin \left (9 d x +9 c \right )-315 \cos \left (8 d x +8 c \right )-180 \sin \left (7 d x +7 c \right )+1008 \sin \left (5 d x +5 c \right )-840 \cos \left (6 d x +6 c \right )-7560 \sin \left (d x +c \right )+1680 \sin \left (3 d x +3 c \right )+1260 \cos \left (4 d x +4 c \right )\right )}{322560 d}\) | \(105\) |
risch | \(\frac {3 a \sin \left (d x +c \right )}{128 d}+\frac {a \sin \left (9 d x +9 c \right )}{2304 d}+\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {a \sin \left (7 d x +7 c \right )}{1792 d}+\frac {a \cos \left (6 d x +6 c \right )}{384 d}-\frac {a \sin \left (5 d x +5 c \right )}{320 d}-\frac {a \cos \left (4 d x +4 c \right )}{256 d}-\frac {a \sin \left (3 d x +3 c \right )}{192 d}-\frac {3 a \cos \left (2 d x +2 c \right )}{128 d}\) | \(134\) |
norman | \(\frac {\frac {32 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {384 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {6976 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}-\frac {384 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {32 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {4 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(205\) |
a/d*(1/9*sin(d*x+c)^9+1/8*sin(d*x+c)^8-2/7*sin(d*x+c)^7-1/3*sin(d*x+c)^6+1 /5*sin(d*x+c)^5+1/4*sin(d*x+c)^4)
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {315 \, a \cos \left (d x + c\right )^{8} - 420 \, a \cos \left (d x + c\right )^{6} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 50 \, a \cos \left (d x + c\right )^{6} + 3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{2520 \, d} \]
1/2520*(315*a*cos(d*x + c)^8 - 420*a*cos(d*x + c)^6 + 8*(35*a*cos(d*x + c) ^8 - 50*a*cos(d*x + c)^6 + 3*a*cos(d*x + c)^4 + 4*a*cos(d*x + c)^2 + 8*a)* sin(d*x + c))/d
Time = 0.91 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {8 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((8*a*sin(c + d*x)**9/(315*d) + 4*a*sin(c + d*x)**7*cos(c + d*x)* *2/(35*d) + a*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) - a*sin(c + d*x)**2*co s(c + d*x)**6/(6*d) - a*cos(c + d*x)**8/(24*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**3*cos(c)**5, True))
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, a \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, a \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \]
1/2520*(280*a*sin(d*x + c)^9 + 315*a*sin(d*x + c)^8 - 720*a*sin(d*x + c)^7 - 840*a*sin(d*x + c)^6 + 504*a*sin(d*x + c)^5 + 630*a*sin(d*x + c)^4)/d
Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.64 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{128 \, d} \]
1/1024*a*cos(8*d*x + 8*c)/d + 1/384*a*cos(6*d*x + 6*c)/d - 1/256*a*cos(4*d *x + 4*c)/d - 3/128*a*cos(2*d*x + 2*c)/d + 1/2304*a*sin(9*d*x + 9*c)/d + 1 /1792*a*sin(7*d*x + 7*c)/d - 1/320*a*sin(5*d*x + 5*c)/d - 1/192*a*sin(3*d* x + 3*c)/d + 3/128*a*sin(d*x + c)/d
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]