3.11 \(\int \log ^2(c x) \, dx\)

Optimal. Leaf size=19 \[ x \log ^2(c x)-2 x \log (c x)+2 x \]

[Out]

2*x - 2*x*Log[c*x] + x*Log[c*x]^2

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Rubi [A]  time = 0.0045393, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2296, 2295} \[ x \log ^2(c x)-2 x \log (c x)+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*x - 2*x*Log[c*x] + x*Log[c*x]^2

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.000852, size = 19, normalized size = 1. \[ x \log ^2(c x)-2 x \log (c x)+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*x - 2*x*Log[c*x] + x*Log[c*x]^2

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Maple [A]  time = 0.036, size = 20, normalized size = 1.1 \begin{align*} 2\,x-2\,x\ln \left ( cx \right ) +x \left ( \ln \left ( cx \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995963, size = 22, normalized size = 1.16 \begin{align*}{\left (\log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 2\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(c*x)^2 - 2*log(c*x) + 2)*x

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Fricas [A]  time = 0.815711, size = 47, normalized size = 2.47 \begin{align*} x \log \left (c x\right )^{2} - 2 \, x \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, algorithm="fricas")

[Out]

x*log(c*x)^2 - 2*x*log(c*x) + 2*x

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Sympy [A]  time = 0.095907, size = 19, normalized size = 1. \begin{align*} x \log{\left (c x \right )}^{2} - 2 x \log{\left (c x \right )} + 2 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(c*x)**2 - 2*x*log(c*x) + 2*x

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Giac [A]  time = 1.09678, size = 26, normalized size = 1.37 \begin{align*} x \log \left (c x\right )^{2} - 2 \, x \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, algorithm="giac")

[Out]

x*log(c*x)^2 - 2*x*log(c*x) + 2*x