\(\int \frac {\log ^2(c x)}{x^2} \, dx\) [13]

Optimal result

Integrand size = 10, antiderivative size = 26 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {2}{x}-\frac {2 \log (c x)}{x}-\frac {\log ^2(c x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, 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 = {2342, 2341} \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {\log ^2(c x)}{x}-\frac {2 \log (c x)}{x}-\frac {2}{x} \]

[In]

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

[Out]

-2/x - (2*Log[c*x])/x - Log[c*x]^2/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 ^2(c x)}{x}+2 \int \frac {\log (c x)}{x^2} \, dx \\ & = -\frac {2}{x}-\frac {2 \log (c x)}{x}-\frac {\log ^2(c x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {2}{x}-\frac {2 \log (c x)}{x}-\frac {\log ^2(c x)}{x} \]

[In]

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

[Out]

-2/x - (2*Log[c*x])/x - Log[c*x]^2/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
norman	\(\frac {-2-\ln \left (x c \right )^{2}-2 \ln \left (x c \right )}{x}\)	\(21\)
parallelrisch	\(\frac {-2-\ln \left (x c \right )^{2}-2 \ln \left (x c \right )}{x}\)	\(21\)
risch	\(-\frac {2}{x}-\frac {2 \ln \left (x c \right )}{x}-\frac {\ln \left (x c \right )^{2}}{x}\)	\(27\)
parts	\(-\frac {\ln \left (x c \right )^{2}}{x}+2 c \left (-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}\right )\)	\(37\)
derivativedivides	\(c \left (-\frac {\ln \left (x c \right )^{2}}{x c}-\frac {2 \ln \left (x c \right )}{x c}-\frac {2}{x c}\right )\)	\(38\)
default	\(c \left (-\frac {\ln \left (x c \right )^{2}}{x c}-\frac {2 \ln \left (x c \right )}{x c}-\frac {2}{x c}\right )\)	\(38\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 2}{x} \]

[In]

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

[Out]

-(log(c*x)^2 + 2*log(c*x) + 2)/x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=- \frac {\log {\left (c x \right )}^{2}}{x} - \frac {2 \log {\left (c x \right )}}{x} - \frac {2}{x} \]

[In]

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

[Out]

-log(c*x)**2/x - 2*log(c*x)/x - 2/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 2}{x} \]

[In]

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

[Out]

-(log(c*x)^2 + 2*log(c*x) + 2)/x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{2}}{x} - \frac {2 \, \log \left (c x\right )}{x} - \frac {2}{x} \]

[In]

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

[Out]

-log(c*x)^2/x - 2*log(c*x)/x - 2/x

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^2(c x)}{x^2} \, dx=-\frac {{\ln \left (c\,x\right )}^2+2\,\ln \left (c\,x\right )+2}{x} \]

[In]

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

[Out]

-(2*log(c*x) + log(c*x)^2 + 2)/x