3.14 \(\int \frac {\log ^2(c x)}{x^3} \, dx\)

Optimal. Leaf size=32 \[ -\frac {\log ^2(c x)}{2 x^2}-\frac {\log (c x)}{2 x^2}-\frac {1}{4 x^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2305, 2304} \[ -\frac {\log ^2(c x)}{2 x^2}-\frac {\log (c x)}{2 x^2}-\frac {1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x]^2/x^3,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.00 \[ -\frac {\log ^2(c x)}{2 x^2}-\frac {\log (c x)}{2 x^2}-\frac {1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x]^2/x^3,x]

[Out]

-1/4*1/x^2 - Log[c*x]/(2*x^2) - Log[c*x]^2/(2*x^2)

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fricas [A]  time = 0.43, size = 21, normalized size = 0.66 \[ -\frac {2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^2/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 26, normalized size = 0.81 \[ -\frac {\log \left (c x\right )^{2}}{2 \, x^{2}} - \frac {\log \left (c x\right )}{2 \, x^{2}} - \frac {1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^2/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 27, normalized size = 0.84 \[ -\frac {\ln \left (c x \right )^{2}}{2 x^{2}}-\frac {\ln \left (c x \right )}{2 x^{2}}-\frac {1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x)^2/x^3,x)

[Out]

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

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maxima [A]  time = 0.50, size = 21, normalized size = 0.66 \[ -\frac {2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x)^2/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.38, size = 21, normalized size = 0.66 \[ -\frac {\frac {{\ln \left (c\,x\right )}^2}{2}+\frac {\ln \left (c\,x\right )}{2}+\frac {1}{4}}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x)^2/x^3,x)

[Out]

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

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sympy [A]  time = 0.13, size = 29, normalized size = 0.91 \[ - \frac {\log {\left (c x \right )}^{2}}{2 x^{2}} - \frac {\log {\left (c x \right )}}{2 x^{2}} - \frac {1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x)**2/x**3,x)

[Out]

-log(c*x)**2/(2*x**2) - log(c*x)/(2*x**2) - 1/(4*x**2)

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