3.27 \(\int x \cos (x) \sin (x) \, dx\)

Optimal. Leaf size=23 \[ -\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

[Out]

-x/4 + (Cos[x]*Sin[x])/4 + (x*Sin[x]^2)/2

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Rubi [A]  time = 0.0127059, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3443, 2635, 8} \[ -\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[x]*Sin[x],x]

[Out]

-x/4 + (Cos[x]*Sin[x])/4 + (x*Sin[x]^2)/2

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x \cos (x) \sin (x) \, dx &=\frac{1}{2} x \sin ^2(x)-\frac{1}{2} \int \sin ^2(x) \, dx\\ &=\frac{1}{4} \cos (x) \sin (x)+\frac{1}{2} x \sin ^2(x)-\frac{\int 1 \, dx}{4}\\ &=-\frac{x}{4}+\frac{1}{4} \cos (x) \sin (x)+\frac{1}{2} x \sin ^2(x)\\ \end{align*}

Mathematica [A]  time = 0.002663, size = 18, normalized size = 0.78 \[ \frac{1}{8} \sin (2 x)-\frac{1}{4} x \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[x]*Sin[x],x]

[Out]

-(x*Cos[2*x])/4 + Sin[2*x]/8

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Maple [A]  time = 0.003, size = 18, normalized size = 0.8 \begin{align*} -{\frac{x \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{4}}+{\frac{x}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)*sin(x),x)

[Out]

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

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Maxima [A]  time = 0.944388, size = 19, normalized size = 0.83 \begin{align*} -\frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*sin(x),x, algorithm="maxima")

[Out]

-1/4*x*cos(2*x) + 1/8*sin(2*x)

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Fricas [A]  time = 0.446078, size = 61, normalized size = 2.65 \begin{align*} -\frac{1}{2} \, x \cos \left (x\right )^{2} + \frac{1}{4} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.322426, size = 24, normalized size = 1.04 \begin{align*} \frac{x \sin ^{2}{\left (x \right )}}{4} - \frac{x \cos ^{2}{\left (x \right )}}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*sin(x),x)

[Out]

x*sin(x)**2/4 - x*cos(x)**2/4 + sin(x)*cos(x)/4

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Giac [A]  time = 1.14303, size = 19, normalized size = 0.83 \begin{align*} -\frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)*sin(x),x, algorithm="giac")

[Out]

-1/4*x*cos(2*x) + 1/8*sin(2*x)