\(\int \frac {\log ^3(c x)}{x^2} \, dx\) [20]

Optimal result

Integrand size = 10, antiderivative size = 37 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \]

[Out]

-6/x-6*ln(c*x)/x-3*ln(c*x)^2/x-ln(c*x)^3/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log ^3(c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {6 \log (c x)}{x}-\frac {6}{x} \]

[In]

Int[Log[c*x]^3/x^2,x]

[Out]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^3(c x)}{x}+3 \int \frac {\log ^2(c x)}{x^2} \, dx \\ & = -\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x}+6 \int \frac {\log (c x)}{x^2} \, dx \\ & = -\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \]

[In]

Integrate[Log[c*x]^3/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78

method	result	size
norman	\(\frac {-6-3 \ln \left (x c \right )^{2}-\ln \left (x c \right )^{3}-6 \ln \left (x c \right )}{x}\)	\(29\)
parallelrisch	\(\frac {-6-3 \ln \left (x c \right )^{2}-\ln \left (x c \right )^{3}-6 \ln \left (x c \right )}{x}\)	\(29\)
risch	\(-\frac {6}{x}-\frac {6 \ln \left (x c \right )}{x}-\frac {3 \ln \left (x c \right )^{2}}{x}-\frac {\ln \left (x c \right )^{3}}{x}\)	\(38\)
parts	\(-\frac {\ln \left (x c \right )^{3}}{x}+3 c \left (-\frac {\ln \left (x c \right )^{2}}{x c}-\frac {2 \ln \left (x c \right )}{x c}-\frac {2}{x c}\right )\)	\(51\)
derivativedivides	\(c \left (-\frac {\ln \left (x c \right )^{3}}{x c}-\frac {3 \ln \left (x c \right )^{2}}{x c}-\frac {6 \ln \left (x c \right )}{x c}-\frac {6}{x c}\right )\)	\(52\)
default	\(c \left (-\frac {\ln \left (x c \right )^{3}}{x c}-\frac {3 \ln \left (x c \right )^{2}}{x c}-\frac {6 \ln \left (x c \right )}{x c}-\frac {6}{x c}\right )\)	\(52\)

[In]

int(ln(x*c)^3/x^2,x,method=_RETURNVERBOSE)

[Out]

(-6-3*ln(x*c)^2-ln(x*c)^3-6*ln(x*c))/x

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3} + 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 6}{x} \]

[In]

integrate(log(c*x)^3/x^2,x, algorithm="fricas")

[Out]

-(log(c*x)^3 + 3*log(c*x)^2 + 6*log(c*x) + 6)/x

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=- \frac {\log {\left (c x \right )}^{3}}{x} - \frac {3 \log {\left (c x \right )}^{2}}{x} - \frac {6 \log {\left (c x \right )}}{x} - \frac {6}{x} \]

[In]

integrate(ln(c*x)**3/x**2,x)

[Out]

-log(c*x)**3/x - 3*log(c*x)**2/x - 6*log(c*x)/x - 6/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3} + 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 6}{x} \]

[In]

integrate(log(c*x)^3/x^2,x, algorithm="maxima")

[Out]

-(log(c*x)^3 + 3*log(c*x)^2 + 6*log(c*x) + 6)/x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3}}{x} - \frac {3 \, \log \left (c x\right )^{2}}{x} - \frac {6 \, \log \left (c x\right )}{x} - \frac {6}{x} \]

[In]

integrate(log(c*x)^3/x^2,x, algorithm="giac")

[Out]

-log(c*x)^3/x - 3*log(c*x)^2/x - 6*log(c*x)/x - 6/x

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {{\ln \left (c\,x\right )}^3+3\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )+6}{x} \]

[In]

int(log(c*x)^3/x^2,x)

[Out]

-(6*log(c*x) + 3*log(c*x)^2 + log(c*x)^3 + 6)/x