\(\int \frac {\log ^3(c x)}{x^3} \, dx\) [21]

Optimal result

Integrand size = 10, antiderivative size = 45 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {3}{8 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2} \]

[Out]

-3/8/x^2-3/4*ln(c*x)/x^2-3/4*ln(c*x)^2/x^2-1/2*ln(c*x)^3/x^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, 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} \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {\log ^3(c x)}{2 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3}{8 x^2} \]

[In]

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

[Out]

-3/(8*x^2) - (3*Log[c*x])/(4*x^2) - (3*Log[c*x]^2)/(4*x^2) - Log[c*x]^3/(2*x^2)

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)}{2 x^2}+\frac {3}{2} \int \frac {\log ^2(c x)}{x^3} \, dx \\ & = -\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2}+\frac {3}{2} \int \frac {\log (c x)}{x^3} \, dx \\ & = -\frac {3}{8 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {3}{8 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2} \]

[In]

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

[Out]

-3/(8*x^2) - (3*Log[c*x])/(4*x^2) - (3*Log[c*x]^2)/(4*x^2) - Log[c*x]^3/(2*x^2)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64

method	result	size
norman	\(\frac {-\frac {3}{8}-\frac {3 \ln \left (x c \right )^{2}}{4}-\frac {\ln \left (x c \right )^{3}}{2}-\frac {3 \ln \left (x c \right )}{4}}{x^{2}}\)	\(29\)
parallelrisch	\(\frac {-3-4 \ln \left (x c \right )^{3}-6 \ln \left (x c \right )^{2}-6 \ln \left (x c \right )}{8 x^{2}}\)	\(30\)
risch	\(-\frac {3}{8 x^{2}}-\frac {3 \ln \left (x c \right )}{4 x^{2}}-\frac {3 \ln \left (x c \right )^{2}}{4 x^{2}}-\frac {\ln \left (x c \right )^{3}}{2 x^{2}}\)	\(38\)
parts	\(-\frac {\ln \left (x c \right )^{3}}{2 x^{2}}+\frac {3 c^{2} \left (-\frac {\ln \left (x c \right )^{2}}{2 x^{2} c^{2}}-\frac {\ln \left (x c \right )}{2 x^{2} c^{2}}-\frac {1}{4 x^{2} c^{2}}\right )}{2}\)	\(53\)
derivativedivides	\(c^{2} \left (-\frac {\ln \left (x c \right )^{3}}{2 x^{2} c^{2}}-\frac {3 \ln \left (x c \right )^{2}}{4 x^{2} c^{2}}-\frac {3 \ln \left (x c \right )}{4 x^{2} c^{2}}-\frac {3}{8 x^{2} c^{2}}\right )\)	\(54\)
default	\(c^{2} \left (-\frac {\ln \left (x c \right )^{3}}{2 x^{2} c^{2}}-\frac {3 \ln \left (x c \right )^{2}}{4 x^{2} c^{2}}-\frac {3 \ln \left (x c \right )}{4 x^{2} c^{2}}-\frac {3}{8 x^{2} c^{2}}\right )\)	\(54\)

[In]

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

[Out]

(-3/8-3/4*ln(x*c)^2-1/2*ln(x*c)^3-3/4*ln(x*c))/x^2

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]

[In]

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

[Out]

-1/8*(4*log(c*x)^3 + 6*log(c*x)^2 + 6*log(c*x) + 3)/x^2

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=- \frac {\log {\left (c x \right )}^{3}}{2 x^{2}} - \frac {3 \log {\left (c x \right )}^{2}}{4 x^{2}} - \frac {3 \log {\left (c x \right )}}{4 x^{2}} - \frac {3}{8 x^{2}} \]

[In]

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

[Out]

-log(c*x)**3/(2*x**2) - 3*log(c*x)**2/(4*x**2) - 3*log(c*x)/(4*x**2) - 3/(8*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]

[In]

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

[Out]

-1/8*(4*log(c*x)^3 + 6*log(c*x)^2 + 6*log(c*x) + 3)/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {\log \left (c x\right )^{3}}{2 \, x^{2}} - \frac {3 \, \log \left (c x\right )^{2}}{4 \, x^{2}} - \frac {3 \, \log \left (c x\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \]

[In]

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

[Out]

-1/2*log(c*x)^3/x^2 - 3/4*log(c*x)^2/x^2 - 3/4*log(c*x)/x^2 - 3/8/x^2

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {\log ^3(c x)}{x^3} \, dx=-\frac {\frac {{\ln \left (c\,x\right )}^3}{2}+\frac {3\,{\ln \left (c\,x\right )}^2}{4}+\frac {3\,\ln \left (c\,x\right )}{4}+\frac {3}{8}}{x^2} \]

[In]

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

[Out]

-((3*log(c*x))/4 + (3*log(c*x)^2)/4 + log(c*x)^3/2 + 3/8)/x^2