3.74.64 \(\int \frac {-9-3 \log (x^3) \log (\log (x^3))}{250 \log (x^3)} \, dx\)

Optimal. Leaf size=10 \[ -\frac {3}{250} x \log \left (\log \left (x^3\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {12, 6741, 6742, 2300, 2178, 2520} \begin {gather*} -\frac {3}{250} x \log \left (\log \left (x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 3*Log[x^3]*Log[Log[x^3]])/(250*Log[x^3]),x]

[Out]

(-3*x*Log[Log[x^3]])/250

Rule 12

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

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 2300

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

Rule 2520

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

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} &=\frac {1}{250} \int \frac {-9-3 \log \left (x^3\right ) \log \left (\log \left (x^3\right )\right )}{\log \left (x^3\right )} \, dx\\ &=\frac {1}{250} \int \frac {3 \left (-3-\log \left (x^3\right ) \log \left (\log \left (x^3\right )\right )\right )}{\log \left (x^3\right )} \, dx\\ &=\frac {3}{250} \int \frac {-3-\log \left (x^3\right ) \log \left (\log \left (x^3\right )\right )}{\log \left (x^3\right )} \, dx\\ &=\frac {3}{250} \int \left (-\frac {3}{\log \left (x^3\right )}-\log \left (\log \left (x^3\right )\right )\right ) \, dx\\ &=-\left (\frac {3}{250} \int \log \left (\log \left (x^3\right )\right ) \, dx\right )-\frac {9}{250} \int \frac {1}{\log \left (x^3\right )} \, dx\\ &=-\frac {3}{250} x \log \left (\log \left (x^3\right )\right )+\frac {9}{250} \int \frac {1}{\log \left (x^3\right )} \, dx-\frac {(3 x) \operatorname {Subst}\left (\int \frac {e^{x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{250 \sqrt [3]{x^3}}\\ &=-\frac {3 x \text {Ei}\left (\frac {\log \left (x^3\right )}{3}\right )}{250 \sqrt [3]{x^3}}-\frac {3}{250} x \log \left (\log \left (x^3\right )\right )+\frac {(3 x) \operatorname {Subst}\left (\int \frac {e^{x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{250 \sqrt [3]{x^3}}\\ &=-\frac {3}{250} x \log \left (\log \left (x^3\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} -\frac {3}{250} x \log \left (\log \left (x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 3*Log[x^3]*Log[Log[x^3]])/(250*Log[x^3]),x]

[Out]

(-3*x*Log[Log[x^3]])/250

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fricas [A]  time = 0.80, size = 8, normalized size = 0.80 \begin {gather*} -\frac {3}{250} \, x \log \left (\log \left (x^{3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/250*(-3*log(x^3)*log(log(x^3))-9)/log(x^3),x, algorithm="fricas")

[Out]

-3/250*x*log(log(x^3))

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giac [A]  time = 0.20, size = 8, normalized size = 0.80 \begin {gather*} -\frac {3}{250} \, x \log \left (\log \left (x^{3}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/250*(-3*log(x^3)*log(log(x^3))-9)/log(x^3),x, algorithm="giac")

[Out]

-3/250*x*log(log(x^3))

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maple [A]  time = 0.02, size = 9, normalized size = 0.90

method	result	size
norman	\(-\frac {3 x \ln \left (\ln \left (x^{3}\right )\right )}{250}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/250*(-3*ln(x^3)*ln(ln(x^3))-9)/ln(x^3),x,method=_RETURNVERBOSE)

[Out]

-3/250*x*ln(ln(x^3))

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maxima [A]  time = 0.47, size = 12, normalized size = 1.20 \begin {gather*} -\frac {3}{250} \, x \log \relax (3) - \frac {3}{250} \, x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/250*(-3*log(x^3)*log(log(x^3))-9)/log(x^3),x, algorithm="maxima")

[Out]

-3/250*x*log(3) - 3/250*x*log(log(x))

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mupad [B]  time = 4.82, size = 8, normalized size = 0.80 \begin {gather*} -\frac {3\,x\,\ln \left (\ln \left (x^3\right )\right )}{250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*log(x^3)*log(log(x^3)))/250 + 9/250)/log(x^3),x)

[Out]

-(3*x*log(log(x^3)))/250

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sympy [A]  time = 0.31, size = 12, normalized size = 1.20 \begin {gather*} - \frac {3 x \log {\left (\log {\left (x^{3} \right )} \right )}}{250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/250*(-3*ln(x**3)*ln(ln(x**3))-9)/ln(x**3),x)

[Out]

-3*x*log(log(x**3))/250

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