3.18 \(\int \log ^3(c x) \, dx\)

Optimal. Leaf size=28 \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]

[Out]

-6*x + 6*x*Log[c*x] - 3*x*Log[c*x]^2 + x*Log[c*x]^3

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Rubi [A]  time = 0.0079641, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2296, 2295} \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6*x + 6*x*Log[c*x] - 3*x*Log[c*x]^2 + x*Log[c*x]^3

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int \log ^3(c x) \, dx &=x \log ^3(c x)-3 \int \log ^2(c x) \, dx\\ &=-3 x \log ^2(c x)+x \log ^3(c x)+6 \int \log (c x) \, dx\\ &=-6 x+6 x \log (c x)-3 x \log ^2(c x)+x \log ^3(c x)\\ \end{align*}

Mathematica [A]  time = 0.0009562, size = 28, normalized size = 1. \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6*x + 6*x*Log[c*x] - 3*x*Log[c*x]^2 + x*Log[c*x]^3

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Maple [A]  time = 0.035, size = 29, normalized size = 1. \begin{align*} -6\,x+6\,x\ln \left ( cx \right ) -3\,x \left ( \ln \left ( cx \right ) \right ) ^{2}+x \left ( \ln \left ( cx \right ) \right ) ^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*x+6*x*ln(c*x)-3*x*ln(c*x)^2+x*ln(c*x)^3

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Maxima [A]  time = 1.02186, size = 32, normalized size = 1.14 \begin{align*}{\left (\log \left (c x\right )^{3} - 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 6\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(c*x)^3 - 3*log(c*x)^2 + 6*log(c*x) - 6)*x

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Fricas [A]  time = 0.747944, size = 70, normalized size = 2.5 \begin{align*} x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x)^3 - 3*x*log(c*x)^2 + 6*x*log(c*x) - 6*x

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Sympy [A]  time = 0.110192, size = 29, normalized size = 1.04 \begin{align*} x \log{\left (c x \right )}^{3} - 3 x \log{\left (c x \right )}^{2} + 6 x \log{\left (c x \right )} - 6 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x)**3 - 3*x*log(c*x)**2 + 6*x*log(c*x) - 6*x

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Giac [A]  time = 1.13001, size = 38, normalized size = 1.36 \begin{align*} x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x)^3 - 3*x*log(c*x)^2 + 6*x*log(c*x) - 6*x