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

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

[Out]

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

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Rubi [A]  time = 0.0331861, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ -\frac{\log ^3(c x)}{2 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3}{8 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0014731, size = 45, normalized size = 1. \[ -\frac{\log ^3(c x)}{2 x^2}-\frac{3 \log ^2(c x)}{4 x^2}-\frac{3 \log (c x)}{4 x^2}-\frac{3}{8 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 38, normalized size = 0.8 \begin{align*} -{\frac{3}{8\,{x}^{2}}}-{\frac{3\,\ln \left ( cx \right ) }{4\,{x}^{2}}}-{\frac{3\, \left ( \ln \left ( cx \right ) \right ) ^{2}}{4\,{x}^{2}}}-{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06371, size = 39, normalized size = 0.87 \begin{align*} -\frac{4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^3/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 = 0.885273, size = 77, normalized size = 1.71 \begin{align*} -\frac{4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.150497, size = 44, normalized size = 0.98 \begin{align*} - \frac{\log{\left (c x \right )}^{3}}{2 x^{2}} - \frac{3 \log{\left (c x \right )}^{2}}{4 x^{2}} - \frac{3 \log{\left (c x \right )}}{4 x^{2}} - \frac{3}{8 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13499, size = 50, normalized size = 1.11 \begin{align*} -\frac{\log \left (c x\right )^{3}}{2 \, x^{2}} - \frac{3 \, \log \left (c x\right )^{2}}{4 \, x^{2}} - \frac{3 \, \log \left (c x\right )}{4 \, x^{2}} - \frac{3}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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