Optimal. Leaf size=32 \[ \frac{1}{4} x^4 \log ^2(c x)-\frac{1}{8} x^4 \log (c x)+\frac{x^4}{32} \]
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Rubi [A] time = 0.0199876, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ \frac{1}{4} x^4 \log ^2(c x)-\frac{1}{8} x^4 \log (c x)+\frac{x^4}{32} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^3 \log ^2(c x) \, dx &=\frac{1}{4} x^4 \log ^2(c x)-\frac{1}{2} \int x^3 \log (c x) \, dx\\ &=\frac{x^4}{32}-\frac{1}{8} x^4 \log (c x)+\frac{1}{4} x^4 \log ^2(c x)\\ \end{align*}
Mathematica [A] time = 0.0012158, size = 32, normalized size = 1. \[ \frac{1}{4} x^4 \log ^2(c x)-\frac{1}{8} x^4 \log (c x)+\frac{x^4}{32} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 27, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{32}}-{\frac{{x}^{4}\ln \left ( cx \right ) }{8}}+{\frac{{x}^{4} \left ( \ln \left ( cx \right ) \right ) ^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975037, size = 28, normalized size = 0.88 \begin{align*} \frac{1}{32} \,{\left (8 \, \log \left (c x\right )^{2} - 4 \, \log \left (c 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 = 0.828518, size = 68, normalized size = 2.12 \begin{align*} \frac{1}{4} \, x^{4} \log \left (c x\right )^{2} - \frac{1}{8} \, x^{4} \log \left (c x\right ) + \frac{1}{32} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.110012, size = 26, normalized size = 0.81 \begin{align*} \frac{x^{4} \log{\left (c x \right )}^{2}}{4} - \frac{x^{4} \log{\left (c x \right )}}{8} + \frac{x^{4}}{32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1193, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{4} \, x^{4} \log \left (c x\right )^{2} - \frac{1}{8} \, x^{4} \log \left (c x\right ) + \frac{1}{32} \, x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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