3.59 \(\int \frac{\log ^2(x)}{x} \, dx\)

Optimal. Leaf size=8 \[ \frac{\log ^3(x)}{3} \]

[Out]

Log[x]^3/3

________________________________________________________________________________________

Rubi [A]  time = 0.0116141, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2302, 30} \[ \frac{\log ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]^2/x,x]

[Out]

Log[x]^3/3

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 30

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

Rubi steps

\begin{align*} \int \frac{\log ^2(x)}{x} \, dx &=\operatorname{Subst}\left (\int x^2 \, dx,x,\log (x)\right )\\ &=\frac{\log ^3(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0009859, size = 8, normalized size = 1. \[ \frac{\log ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]^2/x,x]

[Out]

Log[x]^3/3

________________________________________________________________________________________

Maple [A]  time = 0., size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)^2/x,x)

[Out]

1/3*ln(x)^3

________________________________________________________________________________________

Maxima [A]  time = 0.940217, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \log \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2/x,x, algorithm="maxima")

[Out]

1/3*log(x)^3

________________________________________________________________________________________

Fricas [A]  time = 2.00398, size = 19, normalized size = 2.38 \begin{align*} \frac{1}{3} \, \log \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2/x,x, algorithm="fricas")

[Out]

1/3*log(x)^3

________________________________________________________________________________________

Sympy [A]  time = 0.08174, size = 5, normalized size = 0.62 \begin{align*} \frac{\log{\left (x \right )}^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)**2/x,x)

[Out]

log(x)**3/3

________________________________________________________________________________________

Giac [A]  time = 1.09391, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \log \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)^2/x,x, algorithm="giac")

[Out]

1/3*log(x)^3