### 3.153 $$\int \frac{1}{x \log (e^x)} \, dx$$

Optimal. Leaf size=31 $\frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}-\frac{\log (x)}{x-\log \left (e^x\right )}$

[Out]

-(Log[x]/(x - Log[E^x])) + Log[Log[E^x]]/(x - Log[E^x])

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Rubi [A]  time = 0.0157068, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {2160, 2157, 29} $\frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}-\frac{\log (x)}{x-\log \left (e^x\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*Log[E^x]),x]

[Out]

-(Log[x]/(x - Log[E^x])) + Log[Log[E^x]]/(x - Log[E^x])

Rule 2160

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1
/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 29

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

Rubi steps

\begin{align*} \int \frac{1}{x \log \left (e^x\right )} \, dx &=-\frac{\int \frac{1}{x} \, dx}{x-\log \left (e^x\right )}+\frac{\int \frac{1}{\log \left (e^x\right )} \, dx}{x-\log \left (e^x\right )}\\ &=-\frac{\log (x)}{x-\log \left (e^x\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}\\ &=-\frac{\log (x)}{x-\log \left (e^x\right )}+\frac{\log \left (\log \left (e^x\right )\right )}{x-\log \left (e^x\right )}\\ \end{align*}

Mathematica [A]  time = 0.0083698, size = 21, normalized size = 0.68 $\frac{\log \left (\log \left (e^x\right )\right )-\log (x)}{x-\log \left (e^x\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*Log[E^x]),x]

[Out]

(-Log[x] + Log[Log[E^x]])/(x - Log[E^x])

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Maple [A]  time = 0.012, size = 29, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( \ln \left ({{\rm e}^{x}} \right ) \right ) }{\ln \left ({{\rm e}^{x}} \right ) -x}}+{\frac{\ln \left ( x \right ) }{\ln \left ({{\rm e}^{x}} \right ) -x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/ln(exp(x)),x)

[Out]

-1/(ln(exp(x))-x)*ln(ln(exp(x)))+1/(ln(exp(x))-x)*ln(x)

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Maxima [A]  time = 1.00705, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(exp(x)),x, algorithm="maxima")

[Out]

-1/x

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Fricas [A]  time = 1.88555, size = 8, normalized size = 0.26 \begin{align*} -\frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(exp(x)),x, algorithm="fricas")

[Out]

-1/x

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Sympy [A]  time = 0.071003, size = 3, normalized size = 0.1 \begin{align*} - \frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/ln(exp(x)),x)

[Out]

-1/x

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Giac [A]  time = 1.27182, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/log(exp(x)),x, algorithm="giac")

[Out]

-1/x