3.77.39 \(\int -\frac {1}{3 x} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{6} \left (4-\log (5)+\log \left (\frac {(1+\log (2))^4}{9 x^2}\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.25, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 29} \begin {gather*} -\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{3} \int \frac {1}{x} \, dx\right )\\ &=-\frac {\log (x)}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.25 \begin {gather*} -\frac {\log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Log[x]

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fricas [A]  time = 1.39, size = 4, normalized size = 0.17 \begin {gather*} -\frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(x)

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giac [A]  time = 0.15, size = 5, normalized size = 0.21 \begin {gather*} -\frac {1}{3} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 5, normalized size = 0.21

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*ln(x)

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maxima [A]  time = 0.34, size = 4, normalized size = 0.17 \begin {gather*} -\frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(x)

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mupad [B]  time = 0.09, size = 4, normalized size = 0.17 \begin {gather*} -\frac {\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/3

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sympy [A]  time = 0.05, size = 5, normalized size = 0.21 \begin {gather*} - \frac {\log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3/x,x)

[Out]

-log(x)/3

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