3.14.65 \(\int \frac {-4+6 x+(-4+9 x) \log (x)}{(-2 x+3 x^2) \log (x)} \, dx\)

Optimal. Leaf size=20 \[ \log \left (\frac {1}{5} e^3 (2-3 x) x^2 \log ^2(x)\right ) \]

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Rubi [A]  time = 0.24, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1593, 6742, 72, 2302, 29} \begin {gather*} \log (2-3 x)+2 \log (x)+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 6*x + (-4 + 9*x)*Log[x])/((-2*x + 3*x^2)*Log[x]),x]

[Out]

Log[2 - 3*x] + 2*Log[x] + 2*Log[Log[x]]

Rule 29

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+6 x+(-4+9 x) \log (x)}{x (-2+3 x) \log (x)} \, dx\\ &=\int \left (\frac {-4+9 x}{x (-2+3 x)}+\frac {2}{x \log (x)}\right ) \, dx\\ &=2 \int \frac {1}{x \log (x)} \, dx+\int \frac {-4+9 x}{x (-2+3 x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )+\int \left (\frac {2}{x}+\frac {3}{-2+3 x}\right ) \, dx\\ &=\log (2-3 x)+2 \log (x)+2 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 16, normalized size = 0.80 \begin {gather*} \log (2-3 x)+2 \log (x)+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 6*x + (-4 + 9*x)*Log[x])/((-2*x + 3*x^2)*Log[x]),x]

[Out]

Log[2 - 3*x] + 2*Log[x] + 2*Log[Log[x]]

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fricas [A]  time = 0.64, size = 16, normalized size = 0.80 \begin {gather*} \log \left (3 \, x - 2\right ) + 2 \, \log \relax (x) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-4)*log(x)+6*x-4)/(3*x^2-2*x)/log(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.35, size = 16, normalized size = 0.80 \begin {gather*} \log \left (3 \, x - 2\right ) + 2 \, \log \relax (x) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-4)*log(x)+6*x-4)/(3*x^2-2*x)/log(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.23, size = 17, normalized size = 0.85

method	result	size
default	\(\ln \left (3 x -2\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(17\)
norman	\(\ln \left (3 x -2\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(17\)
risch	\(\ln \left (3 x -2\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x-4)*ln(x)+6*x-4)/(3*x^2-2*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 16, normalized size = 0.80 \begin {gather*} \log \left (3 \, x - 2\right ) + 2 \, \log \relax (x) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-4)*log(x)+6*x-4)/(3*x^2-2*x)/log(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.05, size = 14, normalized size = 0.70 \begin {gather*} \ln \left (x-\frac {2}{3}\right )+2\,\ln \left (\ln \relax (x)\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + log(x)*(9*x - 4) - 4)/(log(x)*(2*x - 3*x^2)),x)

[Out]

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

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sympy [A]  time = 0.14, size = 17, normalized size = 0.85 \begin {gather*} 2 \log {\relax (x )} + \log {\left (x - \frac {2}{3} \right )} + 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-4)*ln(x)+6*x-4)/(3*x**2-2*x)/ln(x),x)

[Out]

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

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