Optimal. Leaf size=37 \[ \frac{8 \text{Ei}(4 \log (c x))}{c^4}-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)} \]
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Rubi [A] time = 0.0520612, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2306, 2309, 2178} \[ \frac{8 \text{Ei}(4 \log (c x))}{c^4}-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int \frac{x^3}{\log ^3(c x)} \, dx &=-\frac{x^4}{2 \log ^2(c x)}+2 \int \frac{x^3}{\log ^2(c x)} \, dx\\ &=-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)}+8 \int \frac{x^3}{\log (c x)} \, dx\\ &=-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)}+\frac{8 \operatorname{Subst}\left (\int \frac{e^{4 x}}{x} \, dx,x,\log (c x)\right )}{c^4}\\ &=\frac{8 \text{Ei}(4 \log (c x))}{c^4}-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)}\\ \end{align*}
Mathematica [A] time = 0.0159373, size = 37, normalized size = 1. \[ \frac{8 \text{Ei}(4 \log (c x))}{c^4}-\frac{x^4}{2 \log ^2(c x)}-\frac{2 x^4}{\log (c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 37, normalized size = 1. \begin{align*} -{\frac{{x}^{4}}{2\, \left ( \ln \left ( cx \right ) \right ) ^{2}}}-2\,{\frac{{x}^{4}}{\ln \left ( cx \right ) }}-8\,{\frac{{\it Ei} \left ( 1,-4\,\ln \left ( cx \right ) \right ) }{{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25461, size = 18, normalized size = 0.49 \begin{align*} -\frac{16 \, \Gamma \left (-2, -4 \, \log \left (c x\right )\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.857464, size = 124, normalized size = 3.35 \begin{align*} -\frac{4 \, c^{4} x^{4} \log \left (c x\right ) + c^{4} x^{4} - 16 \, \log \left (c x\right )^{2} \logintegral \left (c^{4} x^{4}\right )}{2 \, c^{4} \log \left (c x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{- 4 x^{4} \log{\left (c x \right )} - x^{4}}{2 \log{\left (c x \right )}^{2}} + 8 \int \frac{x^{3}}{\log{\left (c x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1246, size = 47, normalized size = 1.27 \begin{align*} -\frac{2 \, x^{4}}{\log \left (c x\right )} - \frac{x^{4}}{2 \, \log \left (c x\right )^{2}} + \frac{8 \,{\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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