\(\int \frac {\log (c x)}{x^2} \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 15 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {1}{x}-\frac {\log (c x)}{x} \]

[Out]

-1/x-ln(c*x)/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log (c x)}{x}-\frac {1}{x} \]

[In]

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

[Out]

-x^(-1) - Log[c*x]/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
]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{x}-\frac {\log (c x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {1}{x}-\frac {\log (c x)}{x} \]

[In]

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

[Out]

-x^(-1) - Log[c*x]/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87

method	result	size
norman	\(\frac {-1-\ln \left (x c \right )}{x}\)	\(13\)
parallelrisch	\(\frac {-1-\ln \left (x c \right )}{x}\)	\(13\)
risch	\(-\frac {1}{x}-\frac {\ln \left (x c \right )}{x}\)	\(16\)
parts	\(-\frac {1}{x}-\frac {\ln \left (x c \right )}{x}\)	\(16\)
derivativedivides	\(c \left (-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}\right )\)	\(24\)
default	\(c \left (-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}\right )\)	\(24\)

[In]

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

[Out]

(-1-ln(x*c))/x

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right ) + 1}{x} \]

[In]

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

[Out]

-(log(c*x) + 1)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {\log (c x)}{x^2} \, dx=- \frac {\log {\left (c x \right )}}{x} - \frac {1}{x} \]

[In]

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

[Out]

-log(c*x)/x - 1/x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]

[In]

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

[Out]

-log(c*x)/x - 1/x

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]

[In]

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

[Out]

-log(c*x)/x - 1/x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\ln \left (c\,x\right )+1}{x} \]

[In]

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

[Out]

-(log(c*x) + 1)/x