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

Optimal. Leaf size=26 \[ -\frac{\log ^2(c x)}{x}-\frac{2 \log (c x)}{x}-\frac{2}{x} \]

[Out]

-2/x - (2*Log[c*x])/x - Log[c*x]^2/x

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Rubi [A]  time = 0.0197428, antiderivative size = 26, 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} \[ -\frac{\log ^2(c x)}{x}-\frac{2 \log (c x)}{x}-\frac{2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/x - (2*Log[c*x])/x - Log[c*x]^2/x

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

Mathematica [A]  time = 0.0010911, size = 26, normalized size = 1. \[ -\frac{\log ^2(c x)}{x}-\frac{2 \log (c x)}{x}-\frac{2}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/x - (2*Log[c*x])/x - Log[c*x]^2/x

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Maple [A]  time = 0.036, size = 27, normalized size = 1. \begin{align*} -2\,{x}^{-1}-2\,{\frac{\ln \left ( cx \right ) }{x}}-{\frac{ \left ( \ln \left ( cx \right ) \right ) ^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966634, size = 26, normalized size = 1. \begin{align*} -\frac{\log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(c*x)^2 + 2*log(c*x) + 2)/x

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Fricas [A]  time = 0.794552, size = 46, normalized size = 1.77 \begin{align*} -\frac{\log \left (c x\right )^{2} + 2 \, \log \left (c x\right ) + 2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(c*x)^2 + 2*log(c*x) + 2)/x

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Sympy [A]  time = 0.111153, size = 20, normalized size = 0.77 \begin{align*} - \frac{\log{\left (c x \right )}^{2}}{x} - \frac{2 \log{\left (c x \right )}}{x} - \frac{2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(c*x)**2/x - 2*log(c*x)/x - 2/x

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Giac [A]  time = 1.11971, size = 35, normalized size = 1.35 \begin{align*} -\frac{\log \left (c x\right )^{2}}{x} - \frac{2 \, \log \left (c x\right )}{x} - \frac{2}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(c*x)^2/x - 2*log(c*x)/x - 2/x