\(\int -\frac {1}{x} \, dx\) [7493]

Optimal result

Integrand size = 5, antiderivative size = 15 \[ \int -\frac {1}{x} \, dx=\frac {2 x-x \log (2 x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.27, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {29} \[ \int -\frac {1}{x} \, dx=-\log (x) \]

[In]

Int[-x^(-1),x]

[Out]

-Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.27 \[ \int -\frac {1}{x} \, dx=-\log (x) \]

[In]

Integrate[-x^(-1),x]

[Out]

-Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.33

method	result	size
default	\(-\ln \left (x \right )\)	\(5\)
norman	\(-\ln \left (x \right )\)	\(5\)
risch	\(-\ln \left (x \right )\)	\(5\)
parallelrisch	\(-\ln \left (x \right )\)	\(5\)

[In]

int(-1/x,x,method=_RETURNVERBOSE)

[Out]

-ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.27 \[ \int -\frac {1}{x} \, dx=-\log \left (x\right ) \]

[In]

integrate(-1/x,x, algorithm="fricas")

[Out]

-log(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.20 \[ \int -\frac {1}{x} \, dx=- \log {\left (x \right )} \]

[In]

integrate(-1/x,x)

[Out]

-log(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.27 \[ \int -\frac {1}{x} \, dx=-\log \left (x\right ) \]

[In]

integrate(-1/x,x, algorithm="maxima")

[Out]

-log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.33 \[ \int -\frac {1}{x} \, dx=-\log \left ({\left | x \right |}\right ) \]

[In]

integrate(-1/x,x, algorithm="giac")

[Out]

-log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.27 \[ \int -\frac {1}{x} \, dx=-\ln \left (x\right ) \]

[In]

int(-1/x,x)

[Out]

-log(x)