### 3.15 $$\int x \sin (x) \, dx$$

Optimal. Leaf size=8 $\sin (x)-x \cos (x)$

[Out]

-(x*Cos[x]) + Sin[x]

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Rubi [A]  time = 0.0085667, antiderivative size = 8, 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, 2637} $\sin (x)-x \cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Sin[x],x]

[Out]

-(x*Cos[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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0020667, size = 8, normalized size = 1. $\sin (x)-x \cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Sin[x],x]

[Out]

-(x*Cos[x]) + Sin[x]

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Maple [A]  time = 0., size = 9, normalized size = 1.1 \begin{align*} -x\cos \left ( x \right ) +\sin \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(x),x)

[Out]

-x*cos(x)+sin(x)

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Maxima [A]  time = 0.933074, size = 11, normalized size = 1.38 \begin{align*} -x \cos \left (x\right ) + \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(x) + sin(x)

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Fricas [A]  time = 1.90965, size = 27, normalized size = 3.38 \begin{align*} -x \cos \left (x\right ) + \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(x) + sin(x)

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Sympy [A]  time = 0.166954, size = 7, normalized size = 0.88 \begin{align*} - x \cos{\left (x \right )} + \sin{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(x),x)

[Out]

-x*cos(x) + sin(x)

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Giac [A]  time = 1.04858, size = 11, normalized size = 1.38 \begin{align*} -x \cos \left (x\right ) + \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(x) + sin(x)