3.87.80 \(\int \frac {-3 x+(3-6 x+24 x^5) \log (x^3)-2 x \log (x^3) \log (\log (x^3))}{\log (x^3)} \, dx\)

Optimal. Leaf size=27 \[ x \left (-x+x \left (-2+\frac {3}{x}+4 x^4-\log \left (\log \left (x^3\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6688, 2310, 2178, 2522} \begin {gather*} 4 x^6-3 x^2-x^2 \log \left (\log \left (x^3\right )\right )+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*x + (3 - 6*x + 24*x^5)*Log[x^3] - 2*x*Log[x^3]*Log[Log[x^3]])/Log[x^3],x]

[Out]

3*x - 3*x^2 + 4*x^6 - x^2*Log[Log[x^3]]

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 2310

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

Rule 2522

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3-6 x+24 x^5-\frac {3 x}{\log \left (x^3\right )}-2 x \log \left (\log \left (x^3\right )\right )\right ) \, dx\\ &=3 x-3 x^2+4 x^6-2 \int x \log \left (\log \left (x^3\right )\right ) \, dx-3 \int \frac {x}{\log \left (x^3\right )} \, dx\\ &=3 x-3 x^2+4 x^6-x^2 \log \left (\log \left (x^3\right )\right )+3 \int \frac {x}{\log \left (x^3\right )} \, dx-\frac {x^2 \operatorname {Subst}\left (\int \frac {e^{2 x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{\left (x^3\right )^{2/3}}\\ &=3 x-3 x^2+4 x^6-\frac {x^2 \text {Ei}\left (\frac {2 \log \left (x^3\right )}{3}\right )}{\left (x^3\right )^{2/3}}-x^2 \log \left (\log \left (x^3\right )\right )+\frac {x^2 \operatorname {Subst}\left (\int \frac {e^{2 x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{\left (x^3\right )^{2/3}}\\ &=3 x-3 x^2+4 x^6-x^2 \log \left (\log \left (x^3\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 24, normalized size = 0.89 \begin {gather*} 3 x-3 x^2+4 x^6-x^2 \log \left (\log \left (x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x + (3 - 6*x + 24*x^5)*Log[x^3] - 2*x*Log[x^3]*Log[Log[x^3]])/Log[x^3],x]

[Out]

3*x - 3*x^2 + 4*x^6 - x^2*Log[Log[x^3]]

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fricas [A]  time = 1.01, size = 24, normalized size = 0.89 \begin {gather*} 4 \, x^{6} - x^{2} \log \left (\log \left (x^{3}\right )\right ) - 3 \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x^3)*log(log(x^3))+(24*x^5-6*x+3)*log(x^3)-3*x)/log(x^3),x, algorithm="fricas")

[Out]

4*x^6 - x^2*log(log(x^3)) - 3*x^2 + 3*x

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giac [A]  time = 0.21, size = 24, normalized size = 0.89 \begin {gather*} 4 \, x^{6} - x^{2} \log \left (\log \left (x^{3}\right )\right ) - 3 \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x^3)*log(log(x^3))+(24*x^5-6*x+3)*log(x^3)-3*x)/log(x^3),x, algorithm="giac")

[Out]

4*x^6 - x^2*log(log(x^3)) - 3*x^2 + 3*x

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {-2 x \ln \left (x^{3}\right ) \ln \left (\ln \left (x^{3}\right )\right )+\left (24 x^{5}-6 x +3\right ) \ln \left (x^{3}\right )-3 x}{\ln \left (x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x^3)*ln(ln(x^3))+(24*x^5-6*x+3)*ln(x^3)-3*x)/ln(x^3),x)

[Out]

int((-2*x*ln(x^3)*ln(ln(x^3))+(24*x^5-6*x+3)*ln(x^3)-3*x)/ln(x^3),x)

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maxima [A]  time = 0.47, size = 29, normalized size = 1.07 \begin {gather*} 4 \, x^{6} - x^{2} \log \relax (3) - x^{2} \log \left (\log \relax (x)\right ) - 3 \, x^{2} + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x^3)*log(log(x^3))+(24*x^5-6*x+3)*log(x^3)-3*x)/log(x^3),x, algorithm="maxima")

[Out]

4*x^6 - x^2*log(3) - x^2*log(log(x)) - 3*x^2 + 3*x

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mupad [B]  time = 5.73, size = 24, normalized size = 0.89 \begin {gather*} 3\,x-3\,x^2+4\,x^6-x^2\,\ln \left (\ln \left (x^3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - log(x^3)*(24*x^5 - 6*x + 3) + 2*x*log(x^3)*log(log(x^3)))/log(x^3),x)

[Out]

3*x - 3*x^2 + 4*x^6 - x^2*log(log(x^3))

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sympy [A]  time = 0.31, size = 22, normalized size = 0.81 \begin {gather*} 4 x^{6} - x^{2} \log {\left (\log {\left (x^{3} \right )} \right )} - 3 x^{2} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x**3)*ln(ln(x**3))+(24*x**5-6*x+3)*ln(x**3)-3*x)/ln(x**3),x)

[Out]

4*x**6 - x**2*log(log(x**3)) - 3*x**2 + 3*x

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