3.26 $$\int \log ^2(x) \, dx$$

Optimal. Leaf size=15 $2 x+x \log ^2(x)-2 x \log (x)$

[Out]

2*x - 2*x*Log[x] + x*Log[x]^2

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Rubi [A]  time = 0.0033866, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {2296, 2295} $2 x+x \log ^2(x)-2 x \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x]^2,x]

[Out]

2*x - 2*x*Log[x] + x*Log[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(x) \, dx &=x \log ^2(x)-2 \int \log (x) \, dx\\ &=2 x-2 x \log (x)+x \log ^2(x)\\ \end{align*}

Mathematica [A]  time = 0.0009109, size = 15, normalized size = 1. $2 x+x \log ^2(x)-2 x \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x]^2,x]

[Out]

2*x - 2*x*Log[x] + x*Log[x]^2

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Maple [A]  time = 0., size = 16, normalized size = 1.1 \begin{align*} 2\,x-2\,x\ln \left ( x \right ) +x \left ( \ln \left ( x \right ) \right ) ^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)^2,x)

[Out]

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

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Maxima [A]  time = 0.936374, size = 16, normalized size = 1.07 \begin{align*}{\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x)^2 - 2*log(x) + 2)*x

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Fricas [A]  time = 1.75099, size = 42, normalized size = 2.8 \begin{align*} x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)^2 - 2*x*log(x) + 2*x

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Sympy [A]  time = 0.087332, size = 15, normalized size = 1. \begin{align*} x \log{\left (x \right )}^{2} - 2 x \log{\left (x \right )} + 2 x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)**2 - 2*x*log(x) + 2*x

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Giac [A]  time = 1.04614, size = 20, normalized size = 1.33 \begin{align*} x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)^2 - 2*x*log(x) + 2*x