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

Optimal. Leaf size=15 \[ -\frac{\cos ^4(a+b x)}{4 b} \]

[Out]

-Cos[a + b*x]^4/(4*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0193339, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2565, 30} \[ -\frac{\cos ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3*Sin[a + b*x],x]

[Out]

-Cos[a + b*x]^4/(4*b)

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

Mathematica [A]  time = 0.0045508, size = 15, normalized size = 1. \[ -\frac{\cos ^4(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[a + b*x]^4/(4*b)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(b*x+a)^4/b

________________________________________________________________________________________

Maxima [A]  time = 1.00887, size = 18, normalized size = 1.2 \begin{align*} -\frac{\cos \left (b x + a\right )^{4}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(b*x + a)^4/b

________________________________________________________________________________________

Fricas [A]  time = 1.81433, size = 31, normalized size = 2.07 \begin{align*} -\frac{\cos \left (b x + a\right )^{4}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*cos(b*x + a)^4/b

________________________________________________________________________________________

Sympy [A]  time = 1.11814, size = 41, normalized size = 2.73 \begin{align*} \begin{cases} \frac{\sin ^{4}{\left (a + b x \right )}}{4 b} + \frac{\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sin{\left (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(b*x+a),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14808, size = 32, normalized size = 2.13 \begin{align*} -\frac{\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sin(b*x + a)^4 - 2*sin(b*x + a)^2)/b