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

Optimal. Leaf size=15 \[ -\frac{2 \cos ^5(a+b x)}{5 b} \]

[Out]

(-2*Cos[a + b*x]^5)/(5*b)

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Rubi [A]  time = 0.0329056, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4287, 2565, 30} \[ -\frac{2 \cos ^5(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[a + b*x]^5)/(5*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cos ^3(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cos ^4(a+b x) \sin (a+b x) \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int x^4 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 \cos ^5(a+b x)}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0079298, size = 15, normalized size = 1. \[ -\frac{2 \cos ^5(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[a + b*x]^5)/(5*b)

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Maple [B]  time = 0.014, size = 41, normalized size = 2.7 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{4\,b}}-{\frac{\cos \left ( 3\,bx+3\,a \right ) }{8\,b}}-{\frac{\cos \left ( 5\,bx+5\,a \right ) }{40\,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),x)

[Out]

-1/4*cos(b*x+a)/b-1/8*cos(3*b*x+3*a)/b-1/40*cos(5*b*x+5*a)/b

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Maxima [B]  time = 1.12088, size = 46, normalized size = 3.07 \begin{align*} -\frac{\cos \left (5 \, b x + 5 \, a\right ) + 5 \, \cos \left (3 \, b x + 3 \, a\right ) + 10 \, \cos \left (b x + a\right )}{40 \, 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),x, algorithm="maxima")

[Out]

-1/40*(cos(5*b*x + 5*a) + 5*cos(3*b*x + 3*a) + 10*cos(b*x + a))/b

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Fricas [A]  time = 0.491357, size = 31, normalized size = 2.07 \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{5}}{5 \, 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),x, algorithm="fricas")

[Out]

-2/5*cos(b*x + a)^5/b

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Sympy [A]  time = 19.2011, size = 117, normalized size = 7.8 \begin{align*} \begin{cases} - \frac{2 \sin ^{3}{\left (a + b x \right )} \sin{\left (2 a + 2 b x \right )}}{5 b} - \frac{4 \sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )} \cos{\left (2 a + 2 b x \right )}}{5 b} + \frac{\sin{\left (a + b x \right )} \sin{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} - \frac{2 \cos ^{3}{\left (a + b x \right )} \cos{\left (2 a + 2 b x \right )}}{5 b} & \text{for}\: b \neq 0 \\x \sin{\left (2 a \right )} \cos ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \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),x)

[Out]

Piecewise((-2*sin(a + b*x)**3*sin(2*a + 2*b*x)/(5*b) - 4*sin(a + b*x)**2*cos(a + b*x)*cos(2*a + 2*b*x)/(5*b) +
sin(a + b*x)*sin(2*a + 2*b*x)*cos(a + b*x)**2/(5*b) - 2*cos(a + b*x)**3*cos(2*a + 2*b*x)/(5*b), Ne(b, 0)), (x
*sin(2*a)*cos(a)**3, True))

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Giac [B]  time = 1.2158, size = 54, normalized size = 3.6 \begin{align*} -\frac{\cos \left (5 \, b x + 5 \, a\right )}{40 \, b} - \frac{\cos \left (3 \, b x + 3 \, a\right )}{8 \, b} - \frac{\cos \left (b x + a\right )}{4 \, 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),x, algorithm="giac")

[Out]

-1/40*cos(5*b*x + 5*a)/b - 1/8*cos(3*b*x + 3*a)/b - 1/4*cos(b*x + a)/b