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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 2295

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \[ x \log ^3(c x)-3 x \log ^2(c x)+6 x \log (c x)-6 x \]

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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fricas [A]  time = 0.40, size = 28, normalized size = 1.00 \[ x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \]

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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giac [A]  time = 0.20, size = 28, normalized size = 1.00 \[ x \log \left (c x\right )^{3} - 3 \, x \log \left (c x\right )^{2} + 6 \, x \log \left (c x\right ) - 6 \, x \]

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

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maple [A]  time = 0.03, size = 29, normalized size = 1.04 \[ x \ln \left (c x \right )^{3}-3 x \ln \left (c x \right )^{2}+6 x \ln \left (c x \right )-6 x \]

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 = 0.48, size = 24, normalized size = 0.86 \[ {\left (\log \left (c x\right )^{3} - 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 6\right )} x \]

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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mupad [B]  time = 3.56, size = 24, normalized size = 0.86 \[ x\,\left ({\ln \left (c\,x\right )}^3-3\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 29, normalized size = 1.04 \[ x \log {\left (c x \right )}^{3} - 3 x \log {\left (c x \right )}^{2} + 6 x \log {\left (c x \right )} - 6 x \]

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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