\(\int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx\) [522]

Optimal result

Integrand size = 27, antiderivative size = 89 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}+\frac {8 (a+a \sin (c+d x))^7}{7 a^4 d}-\frac {5 (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {(a+a \sin (c+d x))^9}{9 a^6 d} \]

[Out]

-2/3*(a+a*sin(d*x+c))^6/a^3/d+8/7*(a+a*sin(d*x+c))^7/a^4/d-5/8*(a+a*sin(d*x+c))^8/a^5/d+1/9*(a+a*sin(d*x+c))^9
/a^6/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 78} \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac {5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac {8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(-2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) + (8*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - (5*(a + a*Sin[c + d*x])^8)/(8*a
^5*d) + (a + a*Sin[c + d*x])^9/(9*a^6*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-x)^2 x (a+x)^5}{a} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int (a-x)^2 x (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (-4 a^3 (a+x)^5+8 a^2 (a+x)^6-5 a (a+x)^7+(a+x)^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = -\frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}+\frac {8 (a+a \sin (c+d x))^7}{7 a^4 d}-\frac {5 (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {(a+a \sin (c+d x))^9}{9 a^6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.12 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (4662-9576 \cos (2 (c+d x))-2772 \cos (4 (c+d x))+168 \cos (6 (c+d x))+189 \cos (8 (c+d x))+16632 \sin (c+d x)-1344 \sin (3 (c+d x))-2016 \sin (5 (c+d x))-396 \sin (7 (c+d x))+28 \sin (9 (c+d x)))}{64512 d} \]

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(4662 - 9576*Cos[2*(c + d*x)] - 2772*Cos[4*(c + d*x)] + 168*Cos[6*(c + d*x)] + 189*Cos[8*(c + d*x)] + 166
32*Sin[c + d*x] - 1344*Sin[3*(c + d*x)] - 2016*Sin[5*(c + d*x)] - 396*Sin[7*(c + d*x)] + 28*Sin[9*(c + d*x)]))
/(64512*d)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\sin ^{6}\left (d x +c \right )\right )}{6}-\left (\sin ^{5}\left (d x +c \right )\right )+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\sin ^{3}\left (d x +c \right )+\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(87\)
default	\(\frac {a^{3} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {5 \left (\sin ^{6}\left (d x +c \right )\right )}{6}-\left (\sin ^{5}\left (d x +c \right )\right )+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}+\sin ^{3}\left (d x +c \right )+\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(87\)
parallelrisch	\(-\frac {a^{3} \left (-11991+9576 \cos \left (2 d x +2 c \right )-28 \sin \left (9 d x +9 c \right )-189 \cos \left (8 d x +8 c \right )+396 \sin \left (7 d x +7 c \right )+2016 \sin \left (5 d x +5 c \right )-168 \cos \left (6 d x +6 c \right )-16632 \sin \left (d x +c \right )+1344 \sin \left (3 d x +3 c \right )+2772 \cos \left (4 d x +4 c \right )\right )}{64512 d}\)	\(107\)
risch	\(\frac {33 a^{3} \sin \left (d x +c \right )}{128 d}+\frac {a^{3} \sin \left (9 d x +9 c \right )}{2304 d}+\frac {3 a^{3} \cos \left (8 d x +8 c \right )}{1024 d}-\frac {11 a^{3} \sin \left (7 d x +7 c \right )}{1792 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{384 d}-\frac {a^{3} \sin \left (5 d x +5 c \right )}{32 d}-\frac {11 a^{3} \cos \left (4 d x +4 c \right )}{256 d}-\frac {a^{3} \sin \left (3 d x +3 c \right )}{48 d}-\frac {19 a^{3} \cos \left (2 d x +2 c \right )}{128 d}\)	\(152\)
norman	\(\frac {\frac {2 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {72 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\frac {3872 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d}+\frac {72 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\frac {16 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {18 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {18 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {26 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {26 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {46 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {46 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(303\)

[In]

int(cos(d*x+c)^5*sin(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

a^3/d*(1/9*sin(d*x+c)^9+3/8*sin(d*x+c)^8+1/7*sin(d*x+c)^7-5/6*sin(d*x+c)^6-sin(d*x+c)^5+1/4*sin(d*x+c)^4+sin(d
*x+c)^3+1/2*sin(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {189 \, a^{3} \cos \left (d x + c\right )^{8} - 336 \, a^{3} \cos \left (d x + c\right )^{6} + 8 \, {\left (7 \, a^{3} \cos \left (d x + c\right )^{8} - 37 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 16 \, a^{3}\right )} \sin \left (d x + c\right )}{504 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/504*(189*a^3*cos(d*x + c)^8 - 336*a^3*cos(d*x + c)^6 + 8*(7*a^3*cos(d*x + c)^8 - 37*a^3*cos(d*x + c)^6 + 6*a
^3*cos(d*x + c)^4 + 8*a^3*cos(d*x + c)^2 + 16*a^3)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (78) = 156\).

Time = 0.91 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.27 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {8 a^{3} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{2 d} - \frac {a^{3} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a^{3} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((8*a**3*sin(c + d*x)**9/(315*d) + 4*a**3*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 8*a**3*sin(c + d*x
)**7/(35*d) + a**3*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + 4*a**3*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + a**3
*sin(c + d*x)**3*cos(c + d*x)**4/d - a**3*sin(c + d*x)**2*cos(c + d*x)**6/(2*d) - a**3*cos(c + d*x)**8/(8*d) -
a**3*cos(c + d*x)**6/(6*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)*cos(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {56 \, a^{3} \sin \left (d x + c\right )^{9} + 189 \, a^{3} \sin \left (d x + c\right )^{8} + 72 \, a^{3} \sin \left (d x + c\right )^{7} - 420 \, a^{3} \sin \left (d x + c\right )^{6} - 504 \, a^{3} \sin \left (d x + c\right )^{5} + 126 \, a^{3} \sin \left (d x + c\right )^{4} + 504 \, a^{3} \sin \left (d x + c\right )^{3} + 252 \, a^{3} \sin \left (d x + c\right )^{2}}{504 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/504*(56*a^3*sin(d*x + c)^9 + 189*a^3*sin(d*x + c)^8 + 72*a^3*sin(d*x + c)^7 - 420*a^3*sin(d*x + c)^6 - 504*a
^3*sin(d*x + c)^5 + 126*a^3*sin(d*x + c)^4 + 504*a^3*sin(d*x + c)^3 + 252*a^3*sin(d*x + c)^2)/d

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.70 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a^{3} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {11 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {19 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a^{3} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {11 \, a^{3} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{3} \sin \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {33 \, a^{3} \sin \left (d x + c\right )}{128 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

3/1024*a^3*cos(8*d*x + 8*c)/d + 1/384*a^3*cos(6*d*x + 6*c)/d - 11/256*a^3*cos(4*d*x + 4*c)/d - 19/128*a^3*cos(
2*d*x + 2*c)/d + 1/2304*a^3*sin(9*d*x + 9*c)/d - 11/1792*a^3*sin(7*d*x + 7*c)/d - 1/32*a^3*sin(5*d*x + 5*c)/d
- 1/48*a^3*sin(3*d*x + 3*c)/d + 33/128*a^3*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.21 \[ \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a^3\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {5\,a^3\,{\sin \left (c+d\,x\right )}^6}{6}-a^3\,{\sin \left (c+d\,x\right )}^5+\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{4}+a^3\,{\sin \left (c+d\,x\right )}^3+\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]

[In]

int(cos(c + d*x)^5*sin(c + d*x)*(a + a*sin(c + d*x))^3,x)

[Out]

((a^3*sin(c + d*x)^2)/2 + a^3*sin(c + d*x)^3 + (a^3*sin(c + d*x)^4)/4 - a^3*sin(c + d*x)^5 - (5*a^3*sin(c + d*
x)^6)/6 + (a^3*sin(c + d*x)^7)/7 + (3*a^3*sin(c + d*x)^8)/8 + (a^3*sin(c + d*x)^9)/9)/d