\(\int \frac {2+(-2 x^2+324 x^4) \log (x)}{x \log (x)} \, dx\) [5063]

Optimal result

Integrand size = 24, antiderivative size = 16 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=-x^2+81 x^4+\log \left (\log ^2(x)\right ) \]

[Out]

81*x^4-x^2+ln(ln(x)^2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6873, 12, 6820, 2339, 29} \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 x^4-x^2+2 \log (\log (x)) \]

[In]

Int[(2 + (-2*x^2 + 324*x^4)*Log[x])/(x*Log[x]),x]

[Out]

-x^2 + 81*x^4 + 2*Log[Log[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

Rule 6873

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1-x^2 \log (x)+162 x^4 \log (x)\right )}{x \log (x)} \, dx \\ & = 2 \int \frac {1-x^2 \log (x)+162 x^4 \log (x)}{x \log (x)} \, dx \\ & = 2 \int \left (-x+162 x^3+\frac {1}{x \log (x)}\right ) \, dx \\ & = -x^2+81 x^4+2 \int \frac {1}{x \log (x)} \, dx \\ & = -x^2+81 x^4+2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -x^2+81 x^4+2 \log (\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=-x^2+81 x^4+2 \log (\log (x)) \]

[In]

Integrate[(2 + (-2*x^2 + 324*x^4)*Log[x])/(x*Log[x]),x]

[Out]

-x^2 + 81*x^4 + 2*Log[Log[x]]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\)	\(17\)
norman	\(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\)	\(17\)
parallelrisch	\(81 x^{4}-x^{2}+2 \ln \left (\ln \left (x \right )\right )\)	\(17\)
risch	\(81 x^{4}-x^{2}+\frac {1}{324}+2 \ln \left (\ln \left (x \right )\right )\)	\(18\)
parts	\(\frac {\left (162 x^{2}-1\right )^{2}}{324}+2 \ln \left (\ln \left (x \right )\right )\)	\(18\)

[In]

int(((324*x^4-2*x^2)*ln(x)+2)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

81*x^4-x^2+2*ln(ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="fricas")

[Out]

81*x^4 - x^2 + 2*log(log(x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 x^{4} - x^{2} + 2 \log {\left (\log {\left (x \right )} \right )} \]

[In]

integrate(((324*x**4-2*x**2)*ln(x)+2)/x/ln(x),x)

[Out]

81*x**4 - x**2 + 2*log(log(x))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="maxima")

[Out]

81*x^4 - x^2 + 2*log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=81 \, x^{4} - x^{2} + 2 \, \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((324*x^4-2*x^2)*log(x)+2)/x/log(x),x, algorithm="giac")

[Out]

81*x^4 - x^2 + 2*log(log(x))

Mupad [B] (verification not implemented)

Time = 12.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {2+\left (-2 x^2+324 x^4\right ) \log (x)}{x \log (x)} \, dx=2\,\ln \left (\ln \left (x\right )\right )-x^2+81\,x^4 \]

[In]

int(-(log(x)*(2*x^2 - 324*x^4) - 2)/(x*log(x)),x)

[Out]

2*log(log(x)) - x^2 + 81*x^4