\(\int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+(2 x^2-3 x^3) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx\) [8146]

Optimal result

Integrand size = 49, antiderivative size = 25 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=-4+x^2-x^3+\frac {3 x+\log (4)}{\log (3 x \log (3))} \]

[Out]

(3*x+2*ln(2))/ln(3*x*ln(3))+x^2-4-x^3

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6820, 45, 2367, 2334, 2335, 2339, 30} \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=-x^3+x^2+\frac {3 x}{\log (x \log (27))}+\frac {\log (4)}{\log (x \log (27))} \]

[In]

Int[(-3*x - Log[4] + 3*x*Log[3*x*Log[3]] + (2*x^2 - 3*x^3)*Log[3*x*Log[3]]^2)/(x*Log[3*x*Log[3]]^2),x]

[Out]

x^2 - x^3 + (3*x)/Log[x*Log[27]] + Log[4]/Log[x*Log[27]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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 2367

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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 ((2-3 x) x-\frac {3+\frac {\log (4)}{x}}{\log ^2(x \log (27))}+\frac {3}{\log (x \log (27))}\right ) \, dx \\ & = 3 \int \frac {1}{\log (x \log (27))} \, dx+\int (2-3 x) x \, dx-\int \frac {3+\frac {\log (4)}{x}}{\log ^2(x \log (27))} \, dx \\ & = \frac {3 \operatorname {LogIntegral}(x \log (27))}{\log (27)}+\int \left (2 x-3 x^2\right ) \, dx-\int \left (\frac {3}{\log ^2(x \log (27))}+\frac {\log (4)}{x \log ^2(x \log (27))}\right ) \, dx \\ & = x^2-x^3+\frac {3 \operatorname {LogIntegral}(x \log (27))}{\log (27)}-3 \int \frac {1}{\log ^2(x \log (27))} \, dx-\log (4) \int \frac {1}{x \log ^2(x \log (27))} \, dx \\ & = x^2-x^3+\frac {3 x}{\log (x \log (27))}+\frac {3 \operatorname {LogIntegral}(x \log (27))}{\log (27)}-3 \int \frac {1}{\log (x \log (27))} \, dx-\log (4) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x \log (27))\right ) \\ & = x^2-x^3+\frac {3 x}{\log (x \log (27))}+\frac {\log (4)}{\log (x \log (27))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=x^2-x^3+\frac {3 x}{\log (x \log (27))}+\frac {\log (4)}{\log (x \log (27))} \]

[In]

Integrate[(-3*x - Log[4] + 3*x*Log[3*x*Log[3]] + (2*x^2 - 3*x^3)*Log[3*x*Log[3]]^2)/(x*Log[3*x*Log[3]]^2),x]

[Out]

x^2 - x^3 + (3*x)/Log[x*Log[27]] + Log[4]/Log[x*Log[27]]

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08

method	result	size
risch	\(-x^{3}+x^{2}+\frac {3 x +2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}\)	\(27\)
parts	\(-x^{3}+x^{2}+\frac {2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}+\frac {3 x}{\ln \left (3 x \ln \left (3\right )\right )}\)	\(33\)
norman	\(\frac {x^{2} \ln \left (3 x \ln \left (3\right )\right )+3 x -x^{3} \ln \left (3 x \ln \left (3\right )\right )+2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}\)	\(39\)
parallelrisch	\(\frac {x^{2} \ln \left (3 x \ln \left (3\right )\right )+3 x -x^{3} \ln \left (3 x \ln \left (3\right )\right )+2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}\)	\(39\)
derivativedivides	\(x^{2}-x^{3}+\frac {2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (3 x \ln \left (3\right )\right )\right )}{\ln \left (3\right )}-\frac {-\frac {3 x \ln \left (3\right )}{\ln \left (3 x \ln \left (3\right )\right )}-\operatorname {Ei}_{1}\left (-\ln \left (3 x \ln \left (3\right )\right )\right )}{\ln \left (3\right )}\)	\(70\)
default	\(x^{2}-x^{3}+\frac {2 \ln \left (2\right )}{\ln \left (3 x \ln \left (3\right )\right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (3 x \ln \left (3\right )\right )\right )}{\ln \left (3\right )}-\frac {-\frac {3 x \ln \left (3\right )}{\ln \left (3 x \ln \left (3\right )\right )}-\operatorname {Ei}_{1}\left (-\ln \left (3 x \ln \left (3\right )\right )\right )}{\ln \left (3\right )}\)	\(70\)

[In]

int(((-3*x^3+2*x^2)*ln(3*x*ln(3))^2+3*x*ln(3*x*ln(3))-2*ln(2)-3*x)/x/ln(3*x*ln(3))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=-\frac {{\left (x^{3} - x^{2}\right )} \log \left (3 \, x \log \left (3\right )\right ) - 3 \, x - 2 \, \log \left (2\right )}{\log \left (3 \, x \log \left (3\right )\right )} \]

[In]

integrate(((-3*x^3+2*x^2)*log(3*x*log(3))^2+3*x*log(3*x*log(3))-2*log(2)-3*x)/x/log(3*x*log(3))^2,x, algorithm
="fricas")

[Out]

-((x^3 - x^2)*log(3*x*log(3)) - 3*x - 2*log(2))/log(3*x*log(3))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=- x^{3} + x^{2} + \frac {3 x + 2 \log {\left (2 \right )}}{\log {\left (3 x \log {\left (3 \right )} \right )}} \]

[In]

integrate(((-3*x**3+2*x**2)*ln(3*x*ln(3))**2+3*x*ln(3*x*ln(3))-2*ln(2)-3*x)/x/ln(3*x*ln(3))**2,x)

[Out]

-x**3 + x**2 + (3*x + 2*log(2))/log(3*x*log(3))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=-x^{3} + x^{2} + \frac {{\rm Ei}\left (\log \left (3 \, x \log \left (3\right )\right )\right )}{\log \left (3\right )} - \frac {\Gamma \left (-1, -\log \left (3 \, x \log \left (3\right )\right )\right )}{\log \left (3\right )} + \frac {2 \, \log \left (2\right )}{\log \left (3 \, x \log \left (3\right )\right )} \]

[In]

integrate(((-3*x^3+2*x^2)*log(3*x*log(3))^2+3*x*log(3*x*log(3))-2*log(2)-3*x)/x/log(3*x*log(3))^2,x, algorithm
="maxima")

[Out]

-x^3 + x^2 + Ei(log(3*x*log(3)))/log(3) - gamma(-1, -log(3*x*log(3)))/log(3) + 2*log(2)/log(3*x*log(3))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=-x^{3} + x^{2} + \frac {3 \, x + 2 \, \log \left (2\right )}{\log \left (3\right ) + \log \left (x \log \left (3\right )\right )} \]

[In]

integrate(((-3*x^3+2*x^2)*log(3*x*log(3))^2+3*x*log(3*x*log(3))-2*log(2)-3*x)/x/log(3*x*log(3))^2,x, algorithm
="giac")

[Out]

-x^3 + x^2 + (3*x + 2*log(2))/(log(3) + log(x*log(3)))

Mupad [B] (verification not implemented)

Time = 12.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-3 x-\log (4)+3 x \log (3 x \log (3))+\left (2 x^2-3 x^3\right ) \log ^2(3 x \log (3))}{x \log ^2(3 x \log (3))} \, dx=\frac {3\,x+\ln \left (4\right )}{\ln \left (3\,x\,\ln \left (3\right )\right )}+x^2-x^3 \]

[In]

int(-(3*x + 2*log(2) - log(3*x*log(3))^2*(2*x^2 - 3*x^3) - 3*x*log(3*x*log(3)))/(x*log(3*x*log(3))^2),x)

[Out]

(3*x + log(4))/log(3*x*log(3)) + x^2 - x^3