3.1.59 \(\int \frac {\log ^2(x)}{x} \, dx\) [59]

Optimal. Leaf size=8 \[ \frac {\log ^3(x)}{3} \]

[Out]

1/3*ln(x)^3

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2339, 30} \begin {gather*} \frac {\log ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps

\begin {align*} \int \frac {\log ^2(x)}{x} \, dx &=\text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )\\ &=\frac {\log ^3(x)}{3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\log ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]^3/3

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Maple [A]
time = 0.01, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\ln \left (x \right )^{3}}{3}\)	\(7\)
default	\(\frac {\ln \left (x \right )^{3}}{3}\)	\(7\)
norman	\(\frac {\ln \left (x \right )^{3}}{3}\)	\(7\)
risch	\(\frac {\ln \left (x \right )^{3}}{3}\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*ln(x)^3

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Maxima [A]
time = 2.28, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \log \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(x)^3

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Fricas [A]
time = 0.97, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \log \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(x)^3

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Sympy [A]
time = 0.02, size = 5, normalized size = 0.62 \begin {gather*} \frac {\log {\left (x \right )}^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)**3/3

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Giac [A]
time = 0.59, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \log \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(x)^3

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Mupad [B]
time = 0.06, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\ln \left (x\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)^3/3

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