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

Optimal. Leaf size=45 \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

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Rubi [A]  time = 0.0181902, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2305, 2304} \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int x \log ^3(c x) \, dx &=\frac{1}{2} x^2 \log ^3(c x)-\frac{3}{2} \int x \log ^2(c x) \, dx\\ &=-\frac{3}{4} x^2 \log ^2(c x)+\frac{1}{2} x^2 \log ^3(c x)+\frac{3}{2} \int x \log (c x) \, dx\\ &=-\frac{3 x^2}{8}+\frac{3}{4} x^2 \log (c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{1}{2} x^2 \log ^3(c x)\\ \end{align*}

Mathematica [A]  time = 0.0011034, size = 45, normalized size = 1. \[ \frac{1}{2} x^2 \log ^3(c x)-\frac{3}{4} x^2 \log ^2(c x)+\frac{3}{4} x^2 \log (c x)-\frac{3 x^2}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

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Maple [A]  time = 0.033, size = 38, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{2}}{8}}+{\frac{3\,{x}^{2}\ln \left ( cx \right ) }{4}}-{\frac{3\,{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}{4}}+{\frac{{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{3}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*x^2+3/4*x^2*ln(c*x)-3/4*x^2*ln(c*x)^2+1/2*x^2*ln(c*x)^3

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Maxima [A]  time = 1.0054, size = 39, normalized size = 0.87 \begin{align*} \frac{1}{8} \,{\left (4 \, \log \left (c x\right )^{3} - 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 3\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*log(c*x)^3 - 6*log(c*x)^2 + 6*log(c*x) - 3)*x^2

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Fricas [A]  time = 0.823187, size = 95, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac{3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac{3}{4} \, x^{2} \log \left (c x\right ) - \frac{3}{8} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(c*x)^3 - 3/4*x^2*log(c*x)^2 + 3/4*x^2*log(c*x) - 3/8*x^2

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Sympy [A]  time = 0.122258, size = 42, normalized size = 0.93 \begin{align*} \frac{x^{2} \log{\left (c x \right )}^{3}}{2} - \frac{3 x^{2} \log{\left (c x \right )}^{2}}{4} + \frac{3 x^{2} \log{\left (c x \right )}}{4} - \frac{3 x^{2}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*log(c*x)**3/2 - 3*x**2*log(c*x)**2/4 + 3*x**2*log(c*x)/4 - 3*x**2/8

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Giac [A]  time = 1.09851, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac{3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac{3}{4} \, x^{2} \log \left (c x\right ) - \frac{3}{8} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(c*x)^3 - 3/4*x^2*log(c*x)^2 + 3/4*x^2*log(c*x) - 3/8*x^2