\(\int \frac {\log ^2(c x)}{x} \, dx\) [12]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \log ^3(c x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 30} \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \log ^3(c x) \]

[In]

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

[Out]

Log[c*x]^3/3

Rule 30

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

Rule 2339

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^2 \, dx,x,\log (c x)\right ) \\ & = \frac {1}{3} \log ^3(c x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \log ^3(c x) \]

[In]

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

[Out]

Log[c*x]^3/3

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\ln \left (x c \right )^{3}}{3}\)	\(9\)
default	\(\frac {\ln \left (x c \right )^{3}}{3}\)	\(9\)
norman	\(\frac {\ln \left (x c \right )^{3}}{3}\)	\(9\)
risch	\(\frac {\ln \left (x c \right )^{3}}{3}\)	\(9\)
parts	\(\ln \left (x c \right )^{2} \ln \left (x \right )-\ln \left (x c \right ) \ln \left (x \right )^{2}+\frac {\ln \left (x \right )^{3}}{3}\)	\(27\)

[In]

int(ln(x*c)^2/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \, \log \left (c x\right )^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {\log {\left (c x \right )}^{3}}{3} \]

[In]

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

[Out]

log(c*x)**3/3

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \, \log \left (c x\right )^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {1}{3} \, \log \left (c x\right )^{3} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^2(c x)}{x} \, dx=\frac {{\ln \left (c\,x\right )}^3}{3} \]

[In]

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

[Out]

log(c*x)^3/3