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

Optimal. Leaf size=22 \[ -9-\left (-4+e^3\right )^2-x^2-\frac {3}{\log ^2(x)} \]

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Rubi [A]  time = 0.11, antiderivative size = 12, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6688, 2302, 30} \begin {gather*} -x^2-\frac {3}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2 - 3/Log[x]^2

Rule 30

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

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.55 \begin {gather*} -x^2-\frac {3}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^2 - 3/Log[x]^2

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fricas [A]  time = 0.58, size = 16, normalized size = 0.73 \begin {gather*} -\frac {x^{2} \log \relax (x)^{2} + 3}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2*log(x)^2 + 3)/log(x)^2

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giac [A]  time = 0.29, size = 12, normalized size = 0.55 \begin {gather*} -x^{2} - \frac {3}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - 3/log(x)^2

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maple [A]  time = 0.01, size = 13, normalized size = 0.59

method	result	size
default	\(-x^{2}-\frac {3}{\ln \relax (x )^{2}}\)	\(13\)
risch	\(-x^{2}-\frac {3}{\ln \relax (x )^{2}}\)	\(13\)
norman	\(\frac {-3-x^{2} \ln \relax (x )^{2}}{\ln \relax (x )^{2}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2-3/ln(x)^2

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maxima [A]  time = 0.34, size = 12, normalized size = 0.55 \begin {gather*} -x^{2} - \frac {3}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - 3/log(x)^2

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mupad [B]  time = 1.26, size = 12, normalized size = 0.55 \begin {gather*} -\frac {3}{{\ln \relax (x)}^2}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 3/log(x)^2 - x^2

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sympy [A]  time = 0.08, size = 10, normalized size = 0.45 \begin {gather*} - x^{2} - \frac {3}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 - 3/log(x)**2

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