3.91.6 \(\int \frac {4}{3 x} \, dx\)

Optimal. Leaf size=31 \[ e^5+\frac {4}{3} \left (-4-e^4-e^6-\log \left (\frac {4 e^4}{x}\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.19, 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 {4 \log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4/(3*x),x]

[Out]

(4*Log[x])/3

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} &=\frac {4}{3} \int \frac {1}{x} \, dx\\ &=\frac {4 \log (x)}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.19 \begin {gather*} \frac {4 \log (x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4/(3*x),x]

[Out]

(4*Log[x])/3

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fricas [A]  time = 0.52, size = 4, normalized size = 0.13 \begin {gather*} \frac {4}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/3/x,x, algorithm="fricas")

[Out]

4/3*log(x)

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giac [A]  time = 0.12, size = 5, normalized size = 0.16 \begin {gather*} \frac {4}{3} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/3/x,x, algorithm="giac")

[Out]

4/3*log(abs(x))

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maple [A]  time = 0.01, size = 5, normalized size = 0.16

method	result	size
default	\(\frac {4 \ln \relax (x )}{3}\)	\(5\)
norman	\(\frac {4 \ln \relax (x )}{3}\)	\(5\)
risch	\(\frac {4 \ln \relax (x )}{3}\)	\(5\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/3/x,x,method=_RETURNVERBOSE)

[Out]

4/3*ln(x)

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maxima [A]  time = 0.34, size = 4, normalized size = 0.13 \begin {gather*} \frac {4}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/3/x,x, algorithm="maxima")

[Out]

4/3*log(x)

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mupad [B]  time = 0.01, size = 4, normalized size = 0.13 \begin {gather*} \frac {4\,\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/(3*x),x)

[Out]

(4*log(x))/3

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sympy [A]  time = 0.05, size = 5, normalized size = 0.16 \begin {gather*} \frac {4 \log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/3/x,x)

[Out]

4*log(x)/3

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