3.1.26 \(\int \log ^2(x) \, dx\) [26]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2333, 2332} \begin {gather*} 2 x+x \log ^2(x)-2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x]^2,x]

[Out]

2*x - 2*x*Log[x] + x*Log[x]^2

Rule 2332

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

Rule 2333

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]

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*}

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} 2 x-2 x \log (x)+x \log ^2(x) \end {gather*}

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 = 0.00, size = 16, normalized size = 1.07

method	result	size
default	\(2 x -2 x \ln \left (x \right )+x \ln \left (x \right )^{2}\)	\(16\)
norman	\(2 x -2 x \ln \left (x \right )+x \ln \left (x \right )^{2}\)	\(16\)
risch	\(2 x -2 x \ln \left (x \right )+x \ln \left (x \right )^{2}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.33, size = 12, normalized size = 0.80 \begin {gather*} {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x \end {gather*}

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 = 0.55, size = 15, normalized size = 1.00 \begin {gather*} x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x \end {gather*}

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 = 0.03, size = 15, normalized size = 1.00 \begin {gather*} x \log {\left (x \right )}^{2} - 2 x \log {\left (x \right )} + 2 x \end {gather*}

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 = 0.64, size = 15, normalized size = 1.00 \begin {gather*} x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x \end {gather*}

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

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.80 \begin {gather*} x\,\left ({\ln \left (x\right )}^2-2\,\ln \left (x\right )+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)^2,x)

[Out]

x*(log(x)^2 - 2*log(x) + 2)

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