3.57 \(\int \log ^2(x) \, dx\)

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 = 0.00791702, 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 \[ 2 x+x \log ^2(x)-2 x \log (x) \]

Antiderivative was successfully verified.

[In]  Int[Log[x]^2,x]

[Out]

2*x - 2*x*Log[x] + x*Log[x]^2

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Rubi in Sympy [A]  time = 0.51416, size = 15, normalized size = 1. \[ x \log{\left (x \right )}^{2} - 2 x \log{\left (x \right )} + 2 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x)**2,x)

[Out]

x*log(x)**2 - 2*x*log(x) + 2*x

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Mathematica [A]  time = 0.0015164, 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 = 0.025, size = 16, normalized size = 1.1 \[ 2\,x-2\,x\ln \left ( x \right ) +x \left ( \ln \left ( x \right ) \right ) ^{2} \]

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 = 1.38508, size = 16, normalized size = 1.07 \[{\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x \]

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 = 0.258505, size = 20, normalized size = 1.33 \[ x \log \left (x\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, x \]

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 = 0.079342, size = 15, normalized size = 1. \[ x \log{\left (x \right )}^{2} - 2 x \log{\left (x \right )} + 2 x \]

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/XCAS [A]  time = 0.220463, size = 20, normalized size = 1.33 \[ x{\rm ln}\left (x\right )^{2} - 2 \, x{\rm ln}\left (x\right ) + 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*ln(x)^2 - 2*x*ln(x) + 2*x