3.71.28 \(\int \frac {-6 x^3+4 x^3 \log (x^3)}{9 \log ^3(x^3)} \, dx\)

Optimal. Leaf size=13 \[ \frac {x^4}{9 \log ^2\left (x^3\right )} \]

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Rubi [C]  time = 0.32, antiderivative size = 181, normalized size of antiderivative = 13.92, number of steps used = 14, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {12, 2561, 2306, 2310, 2178, 2366, 15, 6482} \begin {gather*} -\frac {64 x \log \left (x^3\right ) \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{243 \sqrt [3]{x^3}}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{81 \left (x^3\right )^{4/3}}-\frac {16 x^4 \left (3-2 \log \left (x^3\right )\right ) \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{243 \left (x^3\right )^{4/3}}+\frac {8 x^4 \left (4 \log \left (x^3\right )+3\right ) \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{243 \left (x^3\right )^{4/3}}+\frac {16 x^4}{81}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}-\frac {2 x^4 \left (4 \log \left (x^3\right )+3\right )}{81 \log \left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*x^3 + 4*x^3*Log[x^3])/(9*Log[x^3]^3),x]

[Out]

(16*x^4)/81 + (8*x^4*ExpIntegralEi[(4*Log[x^3])/3])/(81*(x^3)^(4/3)) - (16*x^4*ExpIntegralEi[(4*Log[x^3])/3]*(
3 - 2*Log[x^3]))/(243*(x^3)^(4/3)) + (x^4*(3 - 2*Log[x^3]))/(27*Log[x^3]^2) + (4*x^4*(3 - 2*Log[x^3]))/(81*Log
[x^3]) - (64*x*ExpIntegralEi[(4*Log[x^3])/3]*Log[x^3])/(243*(x^3)^(1/3)) + (8*x^4*ExpIntegralEi[(4*Log[x^3])/3
]*(3 + 4*Log[x^3]))/(243*(x^3)^(4/3)) - (2*x^4*(3 + 4*Log[x^3]))/(81*Log[x^3])

Rule 12

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 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 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 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 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6482

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {-6 x^3+4 x^3 \log \left (x^3\right )}{\log ^3\left (x^3\right )} \, dx\\ &=\frac {1}{9} \int \frac {x^3 \left (-6+4 \log \left (x^3\right )\right )}{\log ^3\left (x^3\right )} \, dx\\ &=-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {4}{3} \int \left (\frac {8 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{27 \sqrt [3]{x^3}}-\frac {x^3 \left (3+4 \log \left (x^3\right )\right )}{18 \log ^2\left (x^3\right )}\right ) \, dx\\ &=-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}+\frac {2}{27} \int \frac {x^3 \left (3+4 \log \left (x^3\right )\right )}{\log ^2\left (x^3\right )} \, dx-\frac {32}{81} \int \frac {\text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{\sqrt [3]{x^3}} \, dx\\ &=-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3+4 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}-\frac {2 x^4 \left (3+4 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {8}{9} \int \left (\frac {4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{9 \sqrt [3]{x^3}}-\frac {x^3}{3 \log \left (x^3\right )}\right ) \, dx-\frac {(32 x) \int \frac {\text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{x} \, dx}{81 \sqrt [3]{x^3}}\\ &=-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3+4 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}-\frac {2 x^4 \left (3+4 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}+\frac {8}{27} \int \frac {x^3}{\log \left (x^3\right )} \, dx-\frac {32}{81} \int \frac {\text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{\sqrt [3]{x^3}} \, dx-\frac {(32 x) \operatorname {Subst}\left (\int \text {Ei}\left (\frac {4 x}{3}\right ) \, dx,x,\log \left (x^3\right )\right )}{243 \sqrt [3]{x^3}}\\ &=\frac {8 x^4}{81}-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {32 x \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \log \left (x^3\right )}{243 \sqrt [3]{x^3}}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3+4 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}-\frac {2 x^4 \left (3+4 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}+\frac {\left (8 x^4\right ) \operatorname {Subst}\left (\int \frac {e^{4 x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{81 \left (x^3\right )^{4/3}}-\frac {(32 x) \int \frac {\text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{x} \, dx}{81 \sqrt [3]{x^3}}\\ &=\frac {8 x^4}{81}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{81 \left (x^3\right )^{4/3}}-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {32 x \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \log \left (x^3\right )}{243 \sqrt [3]{x^3}}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3+4 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}-\frac {2 x^4 \left (3+4 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {(32 x) \operatorname {Subst}\left (\int \text {Ei}\left (\frac {4 x}{3}\right ) \, dx,x,\log \left (x^3\right )\right )}{243 \sqrt [3]{x^3}}\\ &=\frac {16 x^4}{81}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right )}{81 \left (x^3\right )^{4/3}}-\frac {16 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3-2 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}+\frac {x^4 \left (3-2 \log \left (x^3\right )\right )}{27 \log ^2\left (x^3\right )}+\frac {4 x^4 \left (3-2 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}-\frac {64 x \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \log \left (x^3\right )}{243 \sqrt [3]{x^3}}+\frac {8 x^4 \text {Ei}\left (\frac {4 \log \left (x^3\right )}{3}\right ) \left (3+4 \log \left (x^3\right )\right )}{243 \left (x^3\right )^{4/3}}-\frac {2 x^4 \left (3+4 \log \left (x^3\right )\right )}{81 \log \left (x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} \frac {x^4}{9 \log ^2\left (x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x^3 + 4*x^3*Log[x^3])/(9*Log[x^3]^3),x]

[Out]

x^4/(9*Log[x^3]^2)

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fricas [A]  time = 0.57, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^{4}}{9 \, \log \left (x^{3}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(4*x^3*log(x^3)-6*x^3)/log(x^3)^3,x, algorithm="fricas")

[Out]

1/9*x^4/log(x^3)^2

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giac [A]  time = 0.14, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^{4}}{9 \, \log \left (x^{3}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(4*x^3*log(x^3)-6*x^3)/log(x^3)^3,x, algorithm="giac")

[Out]

1/9*x^4/log(x^3)^2

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maple [A]  time = 0.02, size = 12, normalized size = 0.92

method	result	size
norman	\(\frac {x^{4}}{9 \ln \left (x^{3}\right )^{2}}\)	\(12\)
risch	\(\frac {x^{4}}{9 \ln \left (x^{3}\right )^{2}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(4*x^3*ln(x^3)-6*x^3)/ln(x^3)^3,x,method=_RETURNVERBOSE)

[Out]

1/9/ln(x^3)^2*x^4

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maxima [A]  time = 0.42, size = 9, normalized size = 0.69 \begin {gather*} \frac {x^{4}}{81 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(4*x^3*log(x^3)-6*x^3)/log(x^3)^3,x, algorithm="maxima")

[Out]

1/81*x^4/log(x)^2

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mupad [B]  time = 4.21, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^4}{9\,{\ln \left (x^3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*log(x^3))/9 - (2*x^3)/3)/log(x^3)^3,x)

[Out]

x^4/(9*log(x^3)^2)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.77 \begin {gather*} \frac {x^{4}}{9 \log {\left (x^{3} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(4*x**3*ln(x**3)-6*x**3)/ln(x**3)**3,x)

[Out]

x**4/(9*log(x**3)**2)

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