### 3.293 $$\int \frac{1+x}{\log (x) (x+\log (x))} \, dx$$

Optimal. Leaf size=13 $\text{li}(x)+\log (\log (x))-\log (x+\log (x))$

[Out]

Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]

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Rubi [A]  time = 0.142422, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {6742, 2353, 2298, 2302, 29, 6684} $\text{li}(x)+\log (\log (x))-\log (x+\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x)/(Log[x]*(x + Log[x])),x]

[Out]

Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]

Rule 6742

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

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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 29

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

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

Mathematica [A]  time = 0.0385612, size = 13, normalized size = 1. $\text{li}(x)+\log (\log (x))-\log (x+\log (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\frac{1+x}{\ln \left ( x \right ) \left ( x+\ln \left ( x \right ) \right ) }}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{x \log \left (x\right )}\,{d x} - \log \left (x + \log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.13983, size = 68, normalized size = 5.23 \begin{align*} -\log \left (x + \log \left (x\right )\right ) + \log \left (\log \left (x\right )\right ) + \logintegral \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(x)) + log(log(x)) + log_integral(x)

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Sympy [A]  time = 2.14364, size = 15, normalized size = 1.15 \begin{align*} - \log{\left (x + \log{\left (x \right )} \right )} + \log{\left (\log{\left (x \right )} \right )} + \operatorname{Ei}{\left (\log{\left (x \right )} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(x)) + log(log(x)) + Ei(log(x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{{\left (x + \log \left (x\right )\right )} \log \left (x\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 1)/((x + log(x))*log(x)), x)