3.19.53 \(\int \frac {-1+3 x}{-3 x^2+x \log (x)} \, dx\)

Optimal. Leaf size=21 \[ e+\log \left (\frac {e^5}{x \left (3-\frac {\log (x)}{x}\right )}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 11, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2561, 6684} \begin {gather*} -\log (3 x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x)/(-3*x^2 + x*Log[x]),x]

[Out]

-Log[3*x - Log[x]]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+3 x}{x (-3 x+\log (x))} \, dx\\ &=-\log (3 x-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 11, normalized size = 0.52 \begin {gather*} -\log (3 x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x)/(-3*x^2 + x*Log[x]),x]

[Out]

-Log[3*x - Log[x]]

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fricas [A]  time = 0.51, size = 9, normalized size = 0.43 \begin {gather*} -\log \left (-3 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)/(x*log(x)-3*x^2),x, algorithm="fricas")

[Out]

-log(-3*x + log(x))

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giac [A]  time = 0.25, size = 9, normalized size = 0.43 \begin {gather*} -\log \left (-3 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)/(x*log(x)-3*x^2),x, algorithm="giac")

[Out]

-log(-3*x + log(x))

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maple [A]  time = 0.02, size = 10, normalized size = 0.48

method	result	size
risch	\(-\ln \left (\ln \relax (x )-3 x \right )\)	\(10\)
norman	\(-\ln \left (-\ln \relax (x )+3 x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x-1)/(x*ln(x)-3*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x)-3*x)

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maxima [A]  time = 0.90, size = 9, normalized size = 0.43 \begin {gather*} -\log \left (-3 \, x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)/(x*log(x)-3*x^2),x, algorithm="maxima")

[Out]

-log(-3*x + log(x))

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mupad [B]  time = 1.23, size = 11, normalized size = 0.52 \begin {gather*} -\ln \left (3\,x-\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 1)/(x*log(x) - 3*x^2),x)

[Out]

-log(3*x - log(x))

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sympy [A]  time = 0.10, size = 8, normalized size = 0.38 \begin {gather*} - \log {\left (- 3 x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x-1)/(x*ln(x)-3*x**2),x)

[Out]

-log(-3*x + log(x))

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