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

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Rubi [A]  time = 0.0189641, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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 ^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*}

Mathematica [A]  time = 0.0012215, size = 32, normalized size = 1. \[ -\frac{\log ^2(c x)}{2 x^2}-\frac{\log (c x)}{2 x^2}-\frac{1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 27, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}}-{\frac{\ln \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{2}}{2\,{x}^{2}}} \end{align*}

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

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Maxima [A]  time = 0.985426, size = 28, normalized size = 0.88 \begin{align*} -\frac{2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \end{align*}

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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Fricas [A]  time = 0.805439, size = 57, normalized size = 1.78 \begin{align*} -\frac{2 \, \log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \end{align*}

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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Sympy [A]  time = 0.128722, size = 29, normalized size = 0.91 \begin{align*} - \frac{\log{\left (c x \right )}^{2}}{2 x^{2}} - \frac{\log{\left (c x \right )}}{2 x^{2}} - \frac{1}{4 x^{2}} \end{align*}

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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Giac [A]  time = 1.1034, size = 35, normalized size = 1.09 \begin{align*} -\frac{\log \left (c x\right )^{2}}{2 \, x^{2}} - \frac{\log \left (c x\right )}{2 \, x^{2}} - \frac{1}{4 \, x^{2}} \end{align*}

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