3.21 $$\int x \cos (x) \, dx$$

Optimal. Leaf size=7 $x \sin (x)+\cos (x)$

[Out]

Cos[x] + x*Sin[x]

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Rubi [A]  time = 0.008778, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3296, 2638} $x \sin (x)+\cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Cos[x],x]

[Out]

Cos[x] + x*Sin[x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x \cos (x) \, dx &=x \sin (x)-\int \sin (x) \, dx\\ &=\cos (x)+x \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0017807, size = 7, normalized size = 1. $x \sin (x)+\cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Cos[x],x]

[Out]

Cos[x] + x*Sin[x]

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Maple [A]  time = 0., size = 8, normalized size = 1.1 \begin{align*} \cos \left ( x \right ) +x\sin \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x),x)

[Out]

cos(x)+x*sin(x)

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Maxima [A]  time = 0.938032, size = 9, normalized size = 1.29 \begin{align*} x \sin \left (x\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin(x) + cos(x)

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Fricas [A]  time = 1.90055, size = 26, normalized size = 3.71 \begin{align*} x \sin \left (x\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin(x) + cos(x)

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Sympy [A]  time = 0.164299, size = 7, normalized size = 1. \begin{align*} x \sin{\left (x \right )} + \cos{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x),x)

[Out]

x*sin(x) + cos(x)

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Giac [A]  time = 1.04116, size = 9, normalized size = 1.29 \begin{align*} x \sin \left (x\right ) + \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin(x) + cos(x)