Optimal. Leaf size=24 \[ \frac{x^4}{2}+4 x^4 \log ^2(x)-2 x^4 \log (x) \]
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Rubi [A] time = 0.0228337, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {12, 2305, 2304} \[ \frac{x^4}{2}+4 x^4 \log ^2(x)-2 x^4 \log (x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int 16 x^3 \log ^2(x) \, dx &=16 \int x^3 \log ^2(x) \, dx\\ &=4 x^4 \log ^2(x)-8 \int x^3 \log (x) \, dx\\ &=\frac{x^4}{2}-2 x^4 \log (x)+4 x^4 \log ^2(x)\\ \end{align*}
Mathematica [A] time = 0.0012288, size = 30, normalized size = 1.25 \[ 16 \left (\frac{x^4}{32}+\frac{1}{4} x^4 \log ^2(x)-\frac{1}{8} x^4 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1. \begin{align*}{\frac{{x}^{4}}{2}}-2\,{x}^{4}\ln \left ( x \right ) +4\,{x}^{4} \left ( \ln \left ( x \right ) \right ) ^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04629, size = 23, normalized size = 0.96 \begin{align*} \frac{1}{2} \,{\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78746, size = 55, normalized size = 2.29 \begin{align*} 4 \, x^{4} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (x\right ) + \frac{1}{2} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.105006, size = 22, normalized size = 0.92 \begin{align*} 4 x^{4} \log{\left (x \right )}^{2} - 2 x^{4} \log{\left (x \right )} + \frac{x^{4}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34682, size = 30, normalized size = 1.25 \begin{align*} 4 \, x^{4} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (x\right ) + \frac{1}{2} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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