3.1.97 \(\int \frac {-1+3 x-x \log (x)}{-3 x+x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ -5-x+16 \log ^2(3)-\log (25 (3-\log (x))) \]

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Rubi [A]  time = 0.14, antiderivative size = 13, normalized size of antiderivative = 0.59, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2561, 6742, 2302, 29} \begin {gather*} -x-\log (3-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x - x*Log[x])/(-3*x + x*Log[x]),x]

[Out]

-x - Log[3 - Log[x]]

Rule 29

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

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 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 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 {-1+3 x-x \log (x)}{x (-3+\log (x))} \, dx\\ &=\int \left (-1-\frac {1}{x (-3+\log (x))}\right ) \, dx\\ &=-x-\int \frac {1}{x (-3+\log (x))} \, dx\\ &=-x-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-3+\log (x)\right )\\ &=-x-\log (3-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 0.59 \begin {gather*} -x-\log (3-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[3 - Log[x]]

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fricas [A]  time = 0.63, size = 11, normalized size = 0.50 \begin {gather*} -x - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(log(x) - 3)

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giac [A]  time = 0.40, size = 11, normalized size = 0.50 \begin {gather*} -x - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(log(x) - 3)

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maple [A]  time = 0.02, size = 12, normalized size = 0.55

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 11, normalized size = 0.50 \begin {gather*} -x - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(log(x) - 3)

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mupad [B]  time = 0.29, size = 11, normalized size = 0.50 \begin {gather*} -x-\ln \left (\ln \relax (x)-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(x) - 3*x + 1)/(3*x - x*log(x)),x)

[Out]

- x - log(log(x) - 3)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.36 \begin {gather*} - x - \log {\left (\log {\relax (x )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(log(x) - 3)

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