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

Optimal. Leaf size=89 \[ \frac{(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac{5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac{8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]

[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)

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Rubi [A]  time = 0.0834011, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 77} \[ \frac{(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac{5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac{8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]

Antiderivative was successfully verified.

[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 2836

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*x)/b
)^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]

Rule 12

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

Rule 77

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]))))

Rubi steps

\begin{align*} \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 x (a+x)^5}{a} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int (a-x)^2 x (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{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]  time = 0.658734, size = 100, normalized size = 1.12 \[ \frac{a^3 (16632 \sin (c+d x)-1344 \sin (3 (c+d x))-2016 \sin (5 (c+d x))-396 \sin (7 (c+d x))+28 \sin (9 (c+d x))-9576 \cos (2 (c+d x))-2772 \cos (4 (c+d x))+168 \cos (6 (c+d x))+189 \cos (8 (c+d x))+4662)}{64512 d} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.039, size = 170, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{21}}+{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +3\,{a}^{3} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1/24\, \left ( \cos \left ( dx+c \right ) \right ) ^{6} \right ) +3\,{a}^{3} \left ( -1/7\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2
)*sin(d*x+c))+3*a^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+3*a^3*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/3
5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-1/6*a^3*cos(d*x+c)^6)

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Maxima [A]  time = 1.19729, size = 149, normalized size = 1.67 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.43655, size = 240, normalized size = 2.7 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 21.5253, size = 228, normalized size = 2.56 \begin{align*} \begin{cases} \frac{8 a^{3} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{a^{3} \sin ^{8}{\left (c + d x \right )}}{8 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 ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 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{3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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) + a**3*sin(c + d*x)**8/(8*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)**6*cos(c + d*x)**2/(2*d) + a**3*sin(c + d*x)**5*c
os(c + d*x)**4/(5*d) + 4*a**3*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 3*a**3*sin(c + d*x)**4*cos(c + d*x)**4/(
4*d) + a**3*sin(c + d*x)**3*cos(c + d*x)**4/d - a**3*cos(c + d*x)**6/(6*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*si
n(c)*cos(c)**5, True))

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Giac [A]  time = 1.2792, size = 204, normalized size = 2.29 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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