3.71.41 \(\int \frac {3+9 x-3 x \log (x^3)}{x \log ^2(x^3)} \, dx\)

Optimal. Leaf size=15 \[ 2-\frac {1+3 x}{\log \left (x^3\right )} \]

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Rubi [A]  time = 0.24, antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6741, 12, 6742, 2353, 2297, 2300, 2178, 2302, 30} \begin {gather*} -\frac {3 x}{\log \left (x^3\right )}-\frac {1}{\log \left (x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 9*x - 3*x*Log[x^3])/(x*Log[x^3]^2),x]

[Out]

-Log[x^3]^(-1) - (3*x)/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 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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

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 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 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 \left (1+3 x-x \log \left (x^3\right )\right )}{x \log ^2\left (x^3\right )} \, dx\\ &=3 \int \frac {1+3 x-x \log \left (x^3\right )}{x \log ^2\left (x^3\right )} \, dx\\ &=3 \int \left (\frac {1+3 x}{x \log ^2\left (x^3\right )}-\frac {1}{\log \left (x^3\right )}\right ) \, dx\\ &=3 \int \frac {1+3 x}{x \log ^2\left (x^3\right )} \, dx-3 \int \frac {1}{\log \left (x^3\right )} \, dx\\ &=3 \int \left (\frac {3}{\log ^2\left (x^3\right )}+\frac {1}{x \log ^2\left (x^3\right )}\right ) \, dx-\frac {x \operatorname {Subst}\left (\int \frac {e^{x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{\sqrt [3]{x^3}}\\ &=-\frac {x \text {Ei}\left (\frac {\log \left (x^3\right )}{3}\right )}{\sqrt [3]{x^3}}+3 \int \frac {1}{x \log ^2\left (x^3\right )} \, dx+9 \int \frac {1}{\log ^2\left (x^3\right )} \, dx\\ &=-\frac {x \text {Ei}\left (\frac {\log \left (x^3\right )}{3}\right )}{\sqrt [3]{x^3}}-\frac {3 x}{\log \left (x^3\right )}+3 \int \frac {1}{\log \left (x^3\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (x^3\right )\right )\\ &=-\frac {x \text {Ei}\left (\frac {\log \left (x^3\right )}{3}\right )}{\sqrt [3]{x^3}}-\frac {1}{\log \left (x^3\right )}-\frac {3 x}{\log \left (x^3\right )}+\frac {x \operatorname {Subst}\left (\int \frac {e^{x/3}}{x} \, dx,x,\log \left (x^3\right )\right )}{\sqrt [3]{x^3}}\\ &=-\frac {1}{\log \left (x^3\right )}-\frac {3 x}{\log \left (x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 18, normalized size = 1.20 \begin {gather*} -\frac {1}{\log \left (x^3\right )}-\frac {3 x}{\log \left (x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 9*x - 3*x*Log[x^3])/(x*Log[x^3]^2),x]

[Out]

-Log[x^3]^(-1) - (3*x)/Log[x^3]

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fricas [A]  time = 0.85, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, x + 1}{\log \left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x^3)+9*x+3)/x/log(x^3)^2,x, algorithm="fricas")

[Out]

-(3*x + 1)/log(x^3)

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giac [A]  time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, x + 1}{\log \left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x^3)+9*x+3)/x/log(x^3)^2,x, algorithm="giac")

[Out]

-(3*x + 1)/log(x^3)

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maple [A]  time = 0.02, size = 13, normalized size = 0.87

method	result	size
norman	\(\frac {-3 x -1}{\ln \left (x^{3}\right )}\)	\(13\)
risch	\(-\frac {3 x +1}{\ln \left (x^{3}\right )}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x*ln(x^3)+9*x+3)/x/ln(x^3)^2,x,method=_RETURNVERBOSE)

[Out]

(-3*x-1)/ln(x^3)

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maxima [A]  time = 0.40, size = 11, normalized size = 0.73 \begin {gather*} -\frac {3 \, x + 1}{3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(x^3)+9*x+3)/x/log(x^3)^2,x, algorithm="maxima")

[Out]

-1/3*(3*x + 1)/log(x)

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mupad [B]  time = 4.13, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3\,x+1}{\ln \left (x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x + 1)/log(x^3)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.67 \begin {gather*} \frac {- 3 x - 1}{\log {\left (x^{3} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*ln(x**3)+9*x+3)/x/ln(x**3)**2,x)

[Out]

(-3*x - 1)/log(x**3)

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