3.58 \(\int x (\cos (x)+\sin (x)) \, dx\)

Optimal. Leaf size=14 \[ x \sin (x)+\sin (x)-x \cos (x)+\cos (x) \]

[Out]

Cos[x] - x*Cos[x] + Sin[x] + x*Sin[x]

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Rubi [A]  time = 0.0214771, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {14, 3296, 2638, 2637} \[ x \sin (x)+\sin (x)-x \cos (x)+\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[x*(Cos[x] + Sin[x]),x]

[Out]

Cos[x] - x*Cos[x] + Sin[x] + x*Sin[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

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

Mathematica [A]  time = 0.0045986, size = 14, normalized size = 1. \[ x \sin (x)+\sin (x)-x \cos (x)+\cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(Cos[x] + Sin[x]),x]

[Out]

Cos[x] - x*Cos[x] + Sin[x] + x*Sin[x]

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Maple [A]  time = 0.011, size = 15, normalized size = 1.1 \begin{align*} \cos \left ( x \right ) -x\cos \left ( x \right ) +\sin \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)+sin(x)),x)

[Out]

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

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Maxima [A]  time = 0.938204, size = 19, normalized size = 1.36 \begin{align*} -x \cos \left (x\right ) + x \sin \left (x\right ) + \cos \left (x\right ) + \sin \left (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]

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

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Fricas [A]  time = 1.85592, size = 46, normalized size = 3.29 \begin{align*} -{\left (x - 1\right )} \cos \left (x\right ) +{\left (x + 1\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x - 1)*cos(x) + (x + 1)*sin(x)

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Sympy [A]  time = 0.165906, size = 15, normalized size = 1.07 \begin{align*} x \sin{\left (x \right )} - x \cos{\left (x \right )} + \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)+sin(x)),x)

[Out]

x*sin(x) - x*cos(x) + sin(x) + cos(x)

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Giac [A]  time = 1.06894, size = 19, normalized size = 1.36 \begin{align*} -x \cos \left (x\right ) + x \sin \left (x\right ) + \cos \left (x\right ) + \sin \left (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]

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