3.19 \(\int \frac {\log ^3(c x)}{x} \, dx\)

Optimal. Leaf size=10 \[ \frac {1}{4} \log ^4(c x) \]

[Out]

1/4*ln(c*x)^4

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2302, 30} \[ \frac {1}{4} \log ^4(c x) \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]^3/x,x]

[Out]

Log[c*x]^4/4

Rule 30

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

Rule 2302

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 ^3(c x)}{x} \, dx &=\operatorname {Subst}\left (\int x^3 \, dx,x,\log (c x)\right )\\ &=\frac {1}{4} \log ^4(c x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{4} \log ^4(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]^3/x,x]

[Out]

Log[c*x]^4/4

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fricas [A]  time = 0.44, size = 8, normalized size = 0.80 \[ \frac {1}{4} \, \log \left (c x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^3/x,x, algorithm="fricas")

[Out]

1/4*log(c*x)^4

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giac [A]  time = 0.20, size = 8, normalized size = 0.80 \[ \frac {1}{4} \, \log \left (c x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^3/x,x, algorithm="giac")

[Out]

1/4*log(c*x)^4

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maple [A]  time = 0.02, size = 9, normalized size = 0.90 \[ \frac {\ln \left (c x \right )^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x)^3/x,x)

[Out]

1/4*ln(c*x)^4

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maxima [A]  time = 0.49, size = 8, normalized size = 0.80 \[ \frac {1}{4} \, \log \left (c x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^3/x,x, algorithm="maxima")

[Out]

1/4*log(c*x)^4

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mupad [B]  time = 3.53, size = 8, normalized size = 0.80 \[ \frac {{\ln \left (c\,x\right )}^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x)^3/x,x)

[Out]

log(c*x)^4/4

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sympy [A]  time = 0.09, size = 7, normalized size = 0.70 \[ \frac {\log {\left (c x \right )}^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x)**3/x,x)

[Out]

log(c*x)**4/4

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