3.1.58 \(\int x (\cos (x)+\sin (x)) \, dx\) [58]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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, 3377, 2718, 2717} \begin {gather*} x \sin (x)+\sin (x)-x \cos (x)+\cos (x) \end {gather*}

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 2717

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

Rule 2718

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

Rule 3377

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]

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 = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \cos (x)-x \cos (x)+\sin (x)+x \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(Cos[x] + Sin[x]),x]

[Out]

Cos[x] - x*Cos[x] + Sin[x] + x*Sin[x]

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 15, normalized size = 1.07

method	result	size
default	\(\cos \left (x \right )-x \cos \left (x \right )+\sin \left (x \right )+x \sin \left (x \right )\)	\(15\)
risch	\(\left (1-x \right ) \cos \left (x \right )+\left (1+x \right ) \sin \left (x \right )\)	\(16\)
norman	\(\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-x +2 x \tan \left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+2}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(38\)
meijerg	\(2 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (x \right )}{2 \sqrt {\pi }}\right )+2 \sqrt {\pi }\, \left (-\frac {x \cos \left (x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (x \right )}{2 \sqrt {\pi }}\right )\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(cos(x)+sin(x)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 1.95, size = 14, normalized size = 1.00 \begin {gather*} -x \cos \left (x\right ) + x \sin \left (x\right ) + \cos \left (x\right ) + \sin \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.54, size = 14, normalized size = 1.00 \begin {gather*} -{\left (x - 1\right )} \cos \left (x\right ) + {\left (x + 1\right )} \sin \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 15, normalized size = 1.07 \begin {gather*} x \sin {\left (x \right )} - x \cos {\left (x \right )} + \sin {\left (x \right )} + \cos {\left (x \right )} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 14, normalized size = 1.00 \begin {gather*} -x \cos \left (x\right ) + x \sin \left (x\right ) + \cos \left (x\right ) + \sin \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 14, normalized size = 1.00 \begin {gather*} \cos \left (x\right )+\sin \left (x\right )-x\,\cos \left (x\right )+x\,\sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(cos(x) + sin(x)),x)

[Out]

cos(x) + sin(x) - x*cos(x) + x*sin(x)

________________________________________________________________________________________