3.35.69 \(\int \frac {\log (-x)-\log (2 x)}{x \log ^2(-x)} \, dx\)

Optimal. Leaf size=11 \[ \frac {\log (2 x)}{\log (-x)} \]

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Rubi [A]  time = 0.13, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6742, 2302, 29, 30, 2366} \begin {gather*} \frac {\log (2 x)}{\log (-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Log[-x] - Log[2*x])/(x*Log[-x]^2),x]

[Out]

Log[2*x]/Log[-x]

Rule 29

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

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 \left (\frac {1}{x \log (-x)}-\frac {\log (2 x)}{x \log ^2(-x)}\right ) \, dx\\ &=\int \frac {1}{x \log (-x)} \, dx-\int \frac {\log (2 x)}{x \log ^2(-x)} \, dx\\ &=\frac {\log (2 x)}{\log (-x)}-\int \frac {1}{x \log (-x)} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-x)\right )\\ &=\frac {\log (2 x)}{\log (-x)}+\log (\log (-x))-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-x)\right )\\ &=\frac {\log (2 x)}{\log (-x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 19, normalized size = 1.73 \begin {gather*} -\frac {\log (-x)-\log (2 x)}{\log (-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[-x] - Log[2*x])/Log[-x])

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fricas [A]  time = 1.20, size = 9, normalized size = 0.82 \begin {gather*} \frac {\log \relax (2)}{\log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2)/log(-x)

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giac [A]  time = 0.31, size = 18, normalized size = 1.64 \begin {gather*} \frac {\log \left (2 \, x\right ) - \log \left (-x\right )}{\log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(2*x) - log(-x))/log(-x)

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maple [A]  time = 0.05, size = 12, normalized size = 1.09

method	result	size
norman	\(\frac {\ln \left (2 x \right )}{\ln \left (-x \right )}\)	\(12\)
default	\(\frac {\ln \relax (2)}{\ln \left (-x \right )}+\frac {\ln \relax (x )}{\ln \left (-x \right )}\)	\(20\)
risch	\(-\frac {2 i \left (-\ln \relax (2)+i \pi +i \pi \mathrm {csgn}\left (i x \right )^{2} \left (\mathrm {csgn}\left (i x \right )-1\right )\right )}{2 \pi \mathrm {csgn}\left (i x \right )^{2}-2 \pi \mathrm {csgn}\left (i x \right )^{3}-2 \pi +2 i \ln \relax (x )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(2*x)/ln(-x)

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maxima [A]  time = 0.35, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (2 \, x\right )}{\log \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*x)/log(-x)

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mupad [B]  time = 2.03, size = 11, normalized size = 1.00 \begin {gather*} \frac {\ln \left (2\,x\right )}{\ln \left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-x) - log(2*x))/(x*log(-x)^2),x)

[Out]

log(2*x)/log(-x)

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sympy [C]  time = 0.22, size = 10, normalized size = 0.91 \begin {gather*} \frac {\log {\relax (2 )} + i \pi }{\log {\left (- x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(2*x)+ln(-x))/x/ln(-x)**2,x)

[Out]

(log(2) + I*pi)/log(-x)

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