\(\int \frac {2+(2 x+6 x^2) \log (x)}{x \log (x)} \, dx\) [7059]

Optimal result

Integrand size = 22, antiderivative size = 21 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=2+2 x^2+(1+x)^2-\log (2)+\log \left (\log ^2(x)\right ) \]

[Out]

(1+x)^2+ln(ln(x)^2)+2*x^2+2-ln(2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6820, 2339, 29} \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=3 x^2+2 x+2 \log (\log (x)) \]

[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 2339

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 6820

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

Rubi steps \begin{align*} \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 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = 2 x+3 x^2+2 \log (\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=2 x+3 x^2+2 \log (\log (x)) \]

[In]

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

[Out]

2*x + 3*x^2 + 2*Log[Log[x]]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \left (x\right )\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=3 x^{2} + 2 x + 2 \log {\left (\log {\left (x \right )} \right )} \]

[In]

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

[Out]

3*x**2 + 2*x + 2*log(log(x))

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \left (x\right )\right ) \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=3 \, x^{2} + 2 \, x + 2 \, \log \left (\log \left (x\right )\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 13.74 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {2+\left (2 x+6 x^2\right ) \log (x)}{x \log (x)} \, dx=2\,x+2\,\ln \left (\ln \left (x\right )\right )+3\,x^2 \]

[In]

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

[Out]

2*x + 2*log(log(x)) + 3*x^2