3.4.44 \(\int \frac {-6-3 x-6 \log (x)}{x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=13 \[ \frac {3 \left (1+\frac {2}{x}\right )}{\log (x)} \]

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Rubi [A]  time = 0.32, antiderivative size = 16, normalized size of antiderivative = 1.23, number of steps used = 13, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6741, 12, 6742, 2353, 2306, 2309, 2178, 2302, 30} \begin {gather*} \frac {6}{x \log (x)}+\frac {3}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 3*x - 6*Log[x])/(x^2*Log[x]^2),x]

[Out]

3/Log[x] + 6/(x*Log[x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

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 6741

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

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 {3 (-2-x-2 \log (x))}{x^2 \log ^2(x)} \, dx\\ &=3 \int \frac {-2-x-2 \log (x)}{x^2 \log ^2(x)} \, dx\\ &=3 \int \left (\frac {-2-x}{x^2 \log ^2(x)}-\frac {2}{x^2 \log (x)}\right ) \, dx\\ &=3 \int \frac {-2-x}{x^2 \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log (x)} \, dx\\ &=3 \int \left (-\frac {2}{x^2 \log ^2(x)}-\frac {1}{x \log ^2(x)}\right ) \, dx-6 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=-6 \text {Ei}(-\log (x))-3 \int \frac {1}{x \log ^2(x)} \, dx-6 \int \frac {1}{x^2 \log ^2(x)} \, dx\\ &=-6 \text {Ei}(-\log (x))+\frac {6}{x \log (x)}-3 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+6 \int \frac {1}{x^2 \log (x)} \, dx\\ &=-6 \text {Ei}(-\log (x))+\frac {3}{\log (x)}+\frac {6}{x \log (x)}+6 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {3}{\log (x)}+\frac {6}{x \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.23 \begin {gather*} \frac {3}{\log (x)}+\frac {6}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 3*x - 6*Log[x])/(x^2*Log[x]^2),x]

[Out]

3/Log[x] + 6/(x*Log[x])

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fricas [A]  time = 0.96, size = 12, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x + 2\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)-3*x-6)/x^2/log(x)^2,x, algorithm="fricas")

[Out]

3*(x + 2)/(x*log(x))

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giac [A]  time = 0.31, size = 12, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x + 2\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)-3*x-6)/x^2/log(x)^2,x, algorithm="giac")

[Out]

3*(x + 2)/(x*log(x))

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maple [A]  time = 0.02, size = 13, normalized size = 1.00

method	result	size
risch	\(\frac {6+3 x}{x \ln \relax (x )}\)	\(13\)
norman	\(\frac {6+3 x}{x \ln \relax (x )}\)	\(14\)
default	\(\frac {3}{\ln \relax (x )}+\frac {6}{x \ln \relax (x )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*ln(x)-3*x-6)/x^2/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

3*(2+x)/x/ln(x)

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maxima [C]  time = 0.52, size = 20, normalized size = 1.54 \begin {gather*} \frac {3}{\log \relax (x)} - 6 \, {\rm Ei}\left (-\log \relax (x)\right ) + 6 \, \Gamma \left (-1, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(x)-3*x-6)/x^2/log(x)^2,x, algorithm="maxima")

[Out]

3/log(x) - 6*Ei(-log(x)) + 6*gamma(-1, log(x))

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mupad [B]  time = 0.34, size = 12, normalized size = 0.92 \begin {gather*} \frac {3\,\left (x+2\right )}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 6*log(x) + 6)/(x^2*log(x)^2),x)

[Out]

(3*(x + 2))/(x*log(x))

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sympy [A]  time = 0.08, size = 8, normalized size = 0.62 \begin {gather*} \frac {3 x + 6}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*ln(x)-3*x-6)/x**2/ln(x)**2,x)

[Out]

(3*x + 6)/(x*log(x))

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