3.153 \(\int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx\)

Optimal. Leaf size=31 \[ \frac{8 \cos ^9(a+b x)}{9 b}-\frac{8 \cos ^7(a+b x)}{7 b} \]

[Out]

(-8*Cos[a + b*x]^7)/(7*b) + (8*Cos[a + b*x]^9)/(9*b)

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Rubi [A]  time = 0.0560466, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4287, 2565, 14} \[ \frac{8 \cos ^9(a+b x)}{9 b}-\frac{8 \cos ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]

[Out]

(-8*Cos[a + b*x]^7)/(7*b) + (8*Cos[a + b*x]^9)/(9*b)

Rule 4287

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

Rule 2565

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 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]]

Rubi steps

\begin{align*} \int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac{8 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{8 \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{8 \cos ^7(a+b x)}{7 b}+\frac{8 \cos ^9(a+b x)}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.135808, size = 27, normalized size = 0.87 \[ \frac{4 \cos ^7(a+b x) (7 \cos (2 (a+b x))-11)}{63 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]

[Out]

(4*Cos[a + b*x]^7*(-11 + 7*Cos[2*(a + b*x)]))/(63*b)

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Maple [A]  time = 0.017, size = 55, normalized size = 1.8 \begin{align*} -{\frac{3\,\cos \left ( bx+a \right ) }{16\,b}}-{\frac{\cos \left ( 3\,bx+3\,a \right ) }{12\,b}}+{\frac{3\,\cos \left ( 7\,bx+7\,a \right ) }{224\,b}}+{\frac{\cos \left ( 9\,bx+9\,a \right ) }{288\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3*sin(2*b*x+2*a)^3,x)

[Out]

-3/16*cos(b*x+a)/b-1/12*cos(3*b*x+3*a)/b+3/224*cos(7*b*x+7*a)/b+1/288*cos(9*b*x+9*a)/b

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Maxima [A]  time = 1.10552, size = 63, normalized size = 2.03 \begin{align*} \frac{7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{2016 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="maxima")

[Out]

1/2016*(7*cos(9*b*x + 9*a) + 27*cos(7*b*x + 7*a) - 168*cos(3*b*x + 3*a) - 378*cos(b*x + a))/b

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Fricas [A]  time = 0.498608, size = 62, normalized size = 2. \begin{align*} \frac{8 \,{\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="fricas")

[Out]

8/63*(7*cos(b*x + a)^9 - 9*cos(b*x + a)^7)/b

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3*sin(2*b*x+2*a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.29953, size = 73, normalized size = 2.35 \begin{align*} \frac{\cos \left (9 \, b x + 9 \, a\right )}{288 \, b} + \frac{3 \, \cos \left (7 \, b x + 7 \, a\right )}{224 \, b} - \frac{\cos \left (3 \, b x + 3 \, a\right )}{12 \, b} - \frac{3 \, \cos \left (b x + a\right )}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^3,x, algorithm="giac")

[Out]

1/288*cos(9*b*x + 9*a)/b + 3/224*cos(7*b*x + 7*a)/b - 1/12*cos(3*b*x + 3*a)/b - 3/16*cos(b*x + a)/b