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^2-3/4*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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Rubi [A]  time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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
]

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]

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*}

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Mathematica [A]  time = 0.00, size = 45, normalized size = 1.00 \[ -\frac {\log ^3(c x)}{2 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3}{8 x^2} \]

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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fricas [A]  time = 0.43, size = 29, normalized size = 0.64 \[ -\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]

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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giac [A]  time = 0.22, size = 37, normalized size = 0.82 \[ -\frac {\log \left (c x\right )^{3}}{2 \, x^{2}} - \frac {3 \, \log \left (c x\right )^{2}}{4 \, x^{2}} - \frac {3 \, \log \left (c x\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \]

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

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maple [A]  time = 0.03, size = 38, normalized size = 0.84 \[ -\frac {\ln \left (c x \right )^{3}}{2 x^{2}}-\frac {3 \ln \left (c x \right )^{2}}{4 x^{2}}-\frac {3 \ln \left (c x \right )}{4 x^{2}}-\frac {3}{8 x^{2}} \]

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

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maxima [A]  time = 0.57, size = 29, normalized size = 0.64 \[ -\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]

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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mupad [B]  time = 3.60, size = 29, normalized size = 0.64 \[ -\frac {\frac {{\ln \left (c\,x\right )}^3}{2}+\frac {3\,{\ln \left (c\,x\right )}^2}{4}+\frac {3\,\ln \left (c\,x\right )}{4}+\frac {3}{8}}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 44, normalized size = 0.98 \[ - \frac {\log {\left (c x \right )}^{3}}{2 x^{2}} - \frac {3 \log {\left (c x \right )}^{2}}{4 x^{2}} - \frac {3 \log {\left (c x \right )}}{4 x^{2}} - \frac {3}{8 x^{2}} \]

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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