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

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

[Out]

Log[x]*Sin[x]

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Rubi [A]  time = 0.0332612, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, 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 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 = 0.0209431, size = 5, normalized size = 1. \[ \log (x) \sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Log[x] + Sin[x]/x,x]

[Out]

Log[x]*Sin[x]

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Maple [B]  time = 0.05, size = 19, normalized size = 3.8 \begin{align*} 2\,{\frac{\ln \left ( x \right ) \tan \left ( x/2 \right ) }{1+ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}} \end{align*}

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)

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Maxima [A]  time = 1.16891, size = 7, normalized size = 1.4 \begin{align*} \log \left (x\right ) \sin \left (x\right ) \end{align*}

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)

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Fricas [A]  time = 2.24786, size = 20, normalized size = 4. \begin{align*} \log \left (x\right ) \sin \left (x\right ) \end{align*}

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)

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Sympy [A]  time = 12.8869, size = 5, normalized size = 1. \begin{align*} \log{\left (x \right )} \sin{\left (x \right )} \end{align*}

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)

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Giac [A]  time = 1.19441, size = 7, normalized size = 1.4 \begin{align*} \log \left (x\right ) \sin \left (x\right ) \end{align*}

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)