3.1.38 \(\int x \cos (2 x) \, dx\) [38]

Optimal. Leaf size=18 \[ \frac {1}{4} \cos (2 x)+\frac {1}{2} x \sin (2 x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3377, 2718} \begin {gather*} \frac {1}{2} x \sin (2 x)+\frac {1}{4} \cos (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Cos[2*x],x]

[Out]

Cos[2*x]/4 + (x*Sin[2*x])/2

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{4} \cos (2 x)+\frac {1}{2} x \sin (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[2*x],x]

[Out]

Cos[2*x]/4 + (x*Sin[2*x])/2

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 15, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\cos \left (2 x \right )}{4}+\frac {x \sin \left (2 x \right )}{2}\)	\(15\)
default	\(\frac {\cos \left (2 x \right )}{4}+\frac {x \sin \left (2 x \right )}{2}\)	\(15\)
risch	\(\frac {\cos \left (2 x \right )}{4}+\frac {x \sin \left (2 x \right )}{2}\)	\(15\)
norman	\(\frac {x \tan \left (x \right )+\frac {1}{2}}{1+\tan ^{2}\left (x \right )}\)	\(16\)
meijerg	\(\frac {\sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (2 x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (2 x \right )}{\sqrt {\pi }}\right )}{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(2*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 1.78, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x \sin \left (2 \, x\right ) + \frac {1}{4} \, \cos \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(2*x),x, algorithm="maxima")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)

________________________________________________________________________________________

Fricas [A]
time = 0.75, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x \sin \left (2 \, x\right ) + \frac {1}{4} \, \cos \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(2*x),x, algorithm="fricas")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 14, normalized size = 0.78 \begin {gather*} \frac {x \sin {\left (2 x \right )}}{2} + \frac {\cos {\left (2 x \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(2*x),x)

[Out]

x*sin(2*x)/2 + cos(2*x)/4

________________________________________________________________________________________

Giac [A]
time = 0.99, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x \sin \left (2 \, x\right ) + \frac {1}{4} \, \cos \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(2*x),x, algorithm="giac")

[Out]

1/2*x*sin(2*x) + 1/4*cos(2*x)

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 14, normalized size = 0.78 \begin {gather*} \frac {\cos \left (2\,x\right )}{4}+\frac {x\,\sin \left (2\,x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(2*x),x)

[Out]

cos(2*x)/4 + (x*sin(2*x))/2

________________________________________________________________________________________