\(\int -\frac {1}{3 x} \, dx\) [7639]

Optimal result

Integrand size = 7, antiderivative size = 24 \[ \int -\frac {1}{3 x} \, dx=\frac {1}{6} \left (4-\log (5)+\log \left (\frac {(1+\log (2))^4}{9 x^2}\right )\right ) \]

[Out]

2/3-1/6*ln(5)+1/6*ln(1/9*(1+ln(2))^4/x^2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 29} \[ \int -\frac {1}{3 x} \, dx=-\frac {\log (x)}{3} \]

[In]

Int[-1/3*1/x,x]

[Out]

-1/3*Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

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

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \int \frac {1}{x} \, dx\right ) \\ & = -\frac {\log (x)}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.25 \[ \int -\frac {1}{3 x} \, dx=-\frac {\log (x)}{3} \]

[In]

Integrate[-1/3*1/x,x]

[Out]

-1/3*Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.21

method	result	size
default	\(-\frac {\ln \left (x \right )}{3}\)	\(5\)
norman	\(-\frac {\ln \left (x \right )}{3}\)	\(5\)
risch	\(-\frac {\ln \left (x \right )}{3}\)	\(5\)
parallelrisch	\(-\frac {\ln \left (x \right )}{3}\)	\(5\)

[In]

int(-1/3/x,x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.17 \[ \int -\frac {1}{3 x} \, dx=-\frac {1}{3} \, \log \left (x\right ) \]

[In]

integrate(-1/3/x,x, algorithm="fricas")

[Out]

-1/3*log(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.21 \[ \int -\frac {1}{3 x} \, dx=- \frac {\log {\left (x \right )}}{3} \]

[In]

integrate(-1/3/x,x)

[Out]

-log(x)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.17 \[ \int -\frac {1}{3 x} \, dx=-\frac {1}{3} \, \log \left (x\right ) \]

[In]

integrate(-1/3/x,x, algorithm="maxima")

[Out]

-1/3*log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.21 \[ \int -\frac {1}{3 x} \, dx=-\frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate(-1/3/x,x, algorithm="giac")

[Out]

-1/3*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.17 \[ \int -\frac {1}{3 x} \, dx=-\frac {\ln \left (x\right )}{3} \]

[In]

int(-1/(3*x),x)

[Out]

-log(x)/3