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

Optimal. Leaf size=10 \[ \frac{1}{3} \log ^3(c x) \]

[Out]

Log[c*x]^3/3

________________________________________________________________________________________

Rubi [A]  time = 0.0121408, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2302, 30} \[ \frac{1}{3} \log ^3(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*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(c x)}{x} \, dx &=\operatorname{Subst}\left (\int x^2 \, dx,x,\log (c x)\right )\\ &=\frac{1}{3} \log ^3(c x)\\ \end{align*}

Mathematica [A]  time = 0.0009469, size = 10, normalized size = 1. \[ \frac{1}{3} \log ^3(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*x]^3/3

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 9, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.968349, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (c x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(c*x)^3

________________________________________________________________________________________

Fricas [A]  time = 0.790002, size = 22, normalized size = 2.2 \begin{align*} \frac{1}{3} \, \log \left (c x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(c*x)^3

________________________________________________________________________________________

Sympy [A]  time = 0.086911, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (c x \right )}^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x)**3/3

________________________________________________________________________________________

Giac [A]  time = 1.10985, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (c x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(c*x)^3