3.13.74 \(\int \frac {1-x \log (x)}{x \log (x)} \, dx\)

Optimal. Leaf size=7 \[ -x+\log (\log (x)) \]

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Rubi [A]  time = 0.11, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6688, 2302, 29} \begin {gather*} \log (\log (x))-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x*Log[x])/(x*Log[x]),x]

[Out]

-x + Log[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 6688

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -x+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + Log[Log[x]]

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fricas [A]  time = 0.66, size = 7, normalized size = 1.00 \begin {gather*} -x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(log(x))

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giac [A]  time = 0.82, size = 7, normalized size = 1.00 \begin {gather*} -x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(log(x))

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maple [A]  time = 0.01, size = 8, normalized size = 1.14

method	result	size
default	\(\ln \left (\ln \relax (x )\right )-x\)	\(8\)
norman	\(\ln \left (\ln \relax (x )\right )-x\)	\(8\)
risch	\(\ln \left (\ln \relax (x )\right )-x\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(ln(x))-x

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maxima [A]  time = 0.37, size = 7, normalized size = 1.00 \begin {gather*} -x + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(log(x))

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mupad [B]  time = 0.88, size = 7, normalized size = 1.00 \begin {gather*} \ln \left (\ln \relax (x)\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*log(x) - 1)/(x*log(x)),x)

[Out]

log(log(x)) - x

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sympy [A]  time = 0.08, size = 5, normalized size = 0.71 \begin {gather*} - x + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(log(x))

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