3.1.61 \(\int (\cos (x) \log (x)+\frac {\sin (x)}{x}) \, dx\) [61]

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

[Out]

ln(x)*sin(x)

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Rubi [A]
time = 0.02, 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 = {2717, 2634, 3380} \begin {gather*} \log (x) \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Log[x] + Sin[x]/x,x]

[Out]

Log[x]*Sin[x]

Rule 2634

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 2717

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

Rule 3380

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 {gather*} \begin {aligned} \text {Integral} &=\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 {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \log (x) \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Log[x] + Sin[x]/x,x]

[Out]

Log[x]*Sin[x]

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Maple [A]
time = 0.13, size = 6, normalized size = 1.20

method	result	size
risch	\(\ln \left (x \right ) \sin \left (x \right )\)	\(6\)
parallelrisch	\(\ln \left (x \right ) \sin \left (x \right )\)	\(6\)
norman	\(\frac {2 \ln \left (x \right ) \tan \left (\frac {x}{2}\right )}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*sin(x)

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Maxima [A]
time = 0.38, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \sin \left (x\right ) \end {gather*}

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 = 0.60, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \sin \left (x\right ) \end {gather*}

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 = 4.35, size = 5, normalized size = 1.00 \begin {gather*} \log {\left (x \right )} \sin {\left (x \right )} \end {gather*}

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 = 0.47, size = 5, normalized size = 1.00 \begin {gather*} \log \left (x\right ) \sin \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.21, size = 5, normalized size = 1.00 \begin {gather*} \ln \left (x\right )\,\sin \left (x\right ) \end {gather*}

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)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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