Integrand size = 27, antiderivative size = 81 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {b \sin ^5(c+d x)}{5 d}-\frac {2 b \sin ^7(c+d x)}{7 d}+\frac {b \sin ^9(c+d x)}{9 d} \] Output:
-1/6*a*cos(d*x+c)^6/d+1/8*a*cos(d*x+c)^8/d+1/5*b*sin(d*x+c)^5/d-2/7*b*sin( d*x+c)^7/d+1/9*b*sin(d*x+c)^9/d
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.30 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-7560 a \cos (2 (c+d x))-1260 a \cos (4 (c+d x))+840 a \cos (6 (c+d x))+315 a \cos (8 (c+d x))+7560 b \sin (c+d x)-1680 b \sin (3 (c+d x))-1008 b \sin (5 (c+d x))+180 b \sin (7 (c+d x))+140 b \sin (9 (c+d x))}{322560 d} \] Input:
Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]
Output:
(-7560*a*Cos[2*(c + d*x)] - 1260*a*Cos[4*(c + d*x)] + 840*a*Cos[6*(c + d*x )] + 315*a*Cos[8*(c + d*x)] + 7560*b*Sin[c + d*x] - 1680*b*Sin[3*(c + d*x) ] - 1008*b*Sin[5*(c + d*x)] + 180*b*Sin[7*(c + d*x)] + 140*b*Sin[9*(c + d* x)])/(322560*d)
Time = 0.44 (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.
\(\displaystyle \int \sin ^3(c+d x) \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^5 (a+b \sin (c+d x))dx\) |
\(\Big \downarrow \) 3313 |
\(\displaystyle a \int \cos ^5(c+d x) \sin ^3(c+d x)dx+b \int \cos ^5(c+d x) \sin ^4(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+b \int \cos (c+d x)^5 \sin (c+d x)^4dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+\frac {b \int \sin ^4(c+d x) \left (1-\sin ^2(c+d x)\right )^2d\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+\frac {b \int \left (\sin ^8(c+d x)-2 \sin ^6(c+d x)+\sin ^4(c+d x)\right )d\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^5 \sin (c+d x)^3dx+\frac {b \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {b \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \cos ^5(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {b \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \int \left (\cos ^5(c+d x)-\cos ^7(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {1}{9} \sin ^9(c+d x)-\frac {2}{7} \sin ^7(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d}-\frac {a \left (\frac {1}{6} \cos ^6(c+d x)-\frac {1}{8} \cos ^8(c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^6/6 - Cos[c + d*x]^8/8))/d) + (b*(Sin[c + d*x]^5/5 - (2 *Sin[c + d*x]^7)/7 + Sin[c + d*x]^9/9))/d
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 = 87.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {b \sin \left (d x +c \right )^{9}}{9}+\frac {a \sin \left (d x +c \right )^{8}}{8}-\frac {2 b \sin \left (d x +c \right )^{7}}{7}-\frac {a \sin \left (d x +c \right )^{6}}{3}+\frac {\sin \left (d x +c \right )^{5} b}{5}+\frac {\sin \left (d x +c \right )^{4} a}{4}}{d}\) | \(72\) |
default | \(\frac {\frac {b \sin \left (d x +c \right )^{9}}{9}+\frac {a \sin \left (d x +c \right )^{8}}{8}-\frac {2 b \sin \left (d x +c \right )^{7}}{7}-\frac {a \sin \left (d x +c \right )^{6}}{3}+\frac {\sin \left (d x +c \right )^{5} b}{5}+\frac {\sin \left (d x +c \right )^{4} a}{4}}{d}\) | \(72\) |
parallelrisch | \(\frac {7665 a -1260 a \cos \left (4 d x +4 c \right )-1008 b \sin \left (5 d x +5 c \right )+7560 b \sin \left (d x +c \right )-1680 b \sin \left (3 d x +3 c \right )+840 a \cos \left (6 d x +6 c \right )+315 a \cos \left (8 d x +8 c \right )-7560 a \cos \left (2 d x +2 c \right )+140 b \sin \left (9 d x +9 c \right )+180 b \sin \left (7 d x +7 c \right )}{322560 d}\) | \(115\) |
risch | \(\frac {3 b \sin \left (d x +c \right )}{128 d}+\frac {b \sin \left (9 d x +9 c \right )}{2304 d}+\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {b \sin \left (7 d x +7 c \right )}{1792 d}+\frac {a \cos \left (6 d x +6 c \right )}{384 d}-\frac {b \sin \left (5 d x +5 c \right )}{320 d}-\frac {a \cos \left (4 d x +4 c \right )}{256 d}-\frac {b \sin \left (3 d x +3 c \right )}{192 d}-\frac {3 a \cos \left (2 d x +2 c \right )}{128 d}\) | \(134\) |
norman | \(\frac {\frac {32 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}-\frac {384 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {6976 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{315 d}-\frac {384 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{35 d}+\frac {32 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{5 d}-\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(205\) |
orering | \(\text {Expression too large to display}\) | \(1972\) |
Input:
int(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/9*b*sin(d*x+c)^9+1/8*a*sin(d*x+c)^8-2/7*b*sin(d*x+c)^7-1/3*a*sin(d* x+c)^6+1/5*sin(d*x+c)^5*b+1/4*sin(d*x+c)^4*a)
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {315 \, a \cos \left (d x + c\right )^{8} - 420 \, a \cos \left (d x + c\right )^{6} + 8 \, {\left (35 \, b \cos \left (d x + c\right )^{8} - 50 \, b \cos \left (d x + c\right )^{6} + 3 \, b \cos \left (d x + c\right )^{4} + 4 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right )}{2520 \, d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/2520*(315*a*cos(d*x + c)^8 - 420*a*cos(d*x + c)^6 + 8*(35*b*cos(d*x + c) ^8 - 50*b*cos(d*x + c)^6 + 3*b*cos(d*x + c)^4 + 4*b*cos(d*x + c)^2 + 8*b)* sin(d*x + c))/d
Time = 0.95 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.68 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} \frac {a \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {a \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {8 b \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 b \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {b \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \sin ^{3}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**5*sin(d*x+c)**3*(a+b*sin(d*x+c)),x)
Output:
Piecewise((a*sin(c + d*x)**8/(24*d) + a*sin(c + d*x)**6*cos(c + d*x)**2/(6 *d) + a*sin(c + d*x)**4*cos(c + d*x)**4/(4*d) + 8*b*sin(c + d*x)**9/(315*d ) + 4*b*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + b*sin(c + d*x)**5*cos(c + d*x)**4/(5*d), Ne(d, 0)), (x*(a + b*sin(c))*sin(c)**3*cos(c)**5, True))
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {280 \, b \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, b \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, b \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/2520*(280*b*sin(d*x + c)^9 + 315*a*sin(d*x + c)^8 - 720*b*sin(d*x + c)^7 - 840*a*sin(d*x + c)^6 + 504*b*sin(d*x + c)^5 + 630*a*sin(d*x + c)^4)/d
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {280 \, b \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, b \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, b \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/2520*(280*b*sin(d*x + c)^9 + 315*a*sin(d*x + c)^8 - 720*b*sin(d*x + c)^7 - 840*a*sin(d*x + c)^6 + 504*b*sin(d*x + c)^5 + 630*a*sin(d*x + c)^4)/d
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\frac {b\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {b\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \] Input:
int(cos(c + d*x)^5*sin(c + d*x)^3*(a + b*sin(c + d*x)),x)
Output:
((a*sin(c + d*x)^4)/4 - (a*sin(c + d*x)^6)/3 + (a*sin(c + d*x)^8)/8 + (b*s in(c + d*x)^5)/5 - (2*b*sin(c + d*x)^7)/7 + (b*sin(c + d*x)^9)/9)/d
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right )^{4} \left (280 \sin \left (d x +c \right )^{5} b +315 \sin \left (d x +c \right )^{4} a -720 \sin \left (d x +c \right )^{3} b -840 \sin \left (d x +c \right )^{2} a +504 \sin \left (d x +c \right ) b +630 a \right )}{2520 d} \] Input:
int(cos(d*x+c)^5*sin(d*x+c)^3*(a+b*sin(d*x+c)),x)
Output:
(sin(c + d*x)**4*(280*sin(c + d*x)**5*b + 315*sin(c + d*x)**4*a - 720*sin( c + d*x)**3*b - 840*sin(c + d*x)**2*a + 504*sin(c + d*x)*b + 630*a))/(2520 *d)