\(\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [496]

Optimal result

Integrand size = 27, antiderivative size = 81 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{9 d} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2645, 14, 2644, 276} \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^9(c+d x)}{9 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^6(c+d x)}{6 d} \]

[In]

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

[Out]

-1/6*(a*Cos[c + d*x]^6)/d + (a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^5)/(5*d) - (2*a*Sin[c + d*x]^7)/(7*d) +
(a*Sin[c + d*x]^9)/(9*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[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] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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])

Rule 2913

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = a \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+a \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (-7560 \cos (2 (c+d x))-1260 \cos (4 (c+d x))+840 \cos (6 (c+d x))+315 \cos (8 (c+d x))+7560 \sin (c+d x)-1680 \sin (3 (c+d x))-1008 \sin (5 (c+d x))+180 \sin (7 (c+d x))+140 \sin (9 (c+d x)))}{322560 d} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

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	\(\frac {\frac {32 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {384 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {6976 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}-\frac {384 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {32 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {4 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(205\)

[In]

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

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {315 \, a \cos \left (d x + c\right )^{8} - 420 \, a \cos \left (d x + c\right )^{6} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 50 \, a \cos \left (d x + c\right )^{6} + 3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{2520 \, d} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {8 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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*cos(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))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {280 \, a \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, a \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, a \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.64 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{128 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]

[In]

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

[Out]

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