3.160 \(\int (\cos (x) \log (x)+\frac {\sin (x)}{x}) \, dx\)

Optimal. Leaf size=5 \[ \log (x) \sin (x) \]

[Out]

ln(x)*sin(x)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2637, 2554, 3299} \[ \log (x) \sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Log[x] + Sin[x]/x,x]

[Out]

Log[x]*Sin[x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2637

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int \left (\cos (x) \log (x)+\frac {\sin (x)}{x}\right ) \, dx &=\int \cos (x) \log (x) \, dx+\int \frac {\sin (x)}{x} \, dx\\ &=\log (x) \sin (x)+\text {Si}(x)-\int \frac {\sin (x)}{x} \, dx\\ &=\log (x) \sin (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 5, normalized size = 1.00 \[ \log (x) \sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Log[x] + Sin[x]/x,x]

[Out]

Log[x]*Sin[x]

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 5, normalized size = 1.00 \[ \log \relax (x) \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*sin(x)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 5, normalized size = 1.00 \[ \log \relax (x) \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(x)+sin(x)/x,x, algorithm="giac")

[Out]

log(x)*sin(x)

________________________________________________________________________________________

maple [B]  time = 0.93, size = 19, normalized size = 3.80 \[ \frac {2 \ln \relax (x ) \tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*ln(x)+sin(x)/x,x)

[Out]

2*ln(x)*tan(1/2*x)/(1+tan(1/2*x)^2)

________________________________________________________________________________________

maxima [A]  time = 0.93, size = 5, normalized size = 1.00 \[ \log \relax (x) \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*log(x)+sin(x)/x,x, algorithm="maxima")

[Out]

log(x)*sin(x)

________________________________________________________________________________________

mupad [B]  time = 0.50, size = 5, normalized size = 1.00 \[ \ln \relax (x)\,\sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*log(x) + sin(x)/x,x)

[Out]

log(x)*sin(x)

________________________________________________________________________________________

sympy [A]  time = 13.69, size = 5, normalized size = 1.00 \[ \log {\relax (x )} \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*ln(x)+sin(x)/x,x)

[Out]

log(x)*sin(x)

________________________________________________________________________________________