3.71.59 \(\int \frac {2+(2 x+6 x^2) \log (x)}{x \log (x)} \, dx\)

Optimal. Leaf size=21 \[ 2+2 x^2+(1+x)^2-\log (2)+\log \left (\log ^2(x)\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6688, 2302, 29} \begin {gather*} 3 x^2+2 x+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + (2*x + 6*x^2)*Log[x])/(x*Log[x]),x]

[Out]

2*x + 3*x^2 + 2*Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+6 x+\frac {2}{x \log (x)}\right ) \, dx\\ &=2 x+3 x^2+2 \int \frac {1}{x \log (x)} \, dx\\ &=2 x+3 x^2+2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=2 x+3 x^2+2 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.67 \begin {gather*} 2 x+3 x^2+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + (2*x + 6*x^2)*Log[x])/(x*Log[x]),x]

[Out]

2*x + 3*x^2 + 2*Log[Log[x]]

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fricas [A]  time = 0.53, size = 14, normalized size = 0.67 \begin {gather*} 3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+2*x)*log(x)+2)/x/log(x),x, algorithm="fricas")

[Out]

3*x^2 + 2*x + 2*log(log(x))

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giac [A]  time = 0.20, size = 14, normalized size = 0.67 \begin {gather*} 3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+2*x)*log(x)+2)/x/log(x),x, algorithm="giac")

[Out]

3*x^2 + 2*x + 2*log(log(x))

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maple [A]  time = 0.01, size = 15, normalized size = 0.71

method	result	size
default	\(3 x^{2}+2 x +2 \ln \left (\ln \relax (x )\right )\)	\(15\)
norman	\(3 x^{2}+2 x +2 \ln \left (\ln \relax (x )\right )\)	\(15\)
risch	\(3 x^{2}+2 x +2 \ln \left (\ln \relax (x )\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2+2*x)*ln(x)+2)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

3*x^2+2*x+2*ln(ln(x))

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maxima [A]  time = 0.37, size = 14, normalized size = 0.67 \begin {gather*} 3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+2*x)*log(x)+2)/x/log(x),x, algorithm="maxima")

[Out]

3*x^2 + 2*x + 2*log(log(x))

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mupad [B]  time = 4.17, size = 14, normalized size = 0.67 \begin {gather*} 2\,x+2\,\ln \left (\ln \relax (x)\right )+3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x + 6*x^2) + 2)/(x*log(x)),x)

[Out]

2*x + 2*log(log(x)) + 3*x^2

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sympy [A]  time = 0.10, size = 14, normalized size = 0.67 \begin {gather*} 3 x^{2} + 2 x + 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2+2*x)*ln(x)+2)/x/ln(x),x)

[Out]

3*x**2 + 2*x + 2*log(log(x))

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