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 = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 2178
Rule 2306
Rule 2309
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*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.00 \[ \frac {8 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{2 \log ^2(c x)}-\frac {2 x^4}{\log (c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 1.27 \[ -\frac {4 \, c^{4} x^{4} \log \left (c x\right ) + c^{4} x^{4} - 16 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{4} x^{4}\right )}{2 \, c^{4} \log \left (c x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 35, normalized size = 0.95 \[ -\frac {2 \, x^{4}}{\log \left (c x\right )} - \frac {x^{4}}{2 \, \log \left (c x\right )^{2}} + \frac {8 \, {\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 1.00 \[ -\frac {2 x^{4}}{\ln \left (c x \right )}-\frac {x^{4}}{2 \ln \left (c x \right )^{2}}-\frac {8 \Ei \left (1, -4 \ln \left (c x \right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 13, normalized size = 0.35 \[ -\frac {16 \, \Gamma \left (-2, -4 \, \log \left (c x\right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3}{{\ln \left (c\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- 4 x^{4} \log {\left (c x \right )} - x^{4}}{2 \log {\left (c x \right )}^{2}} + 8 \int \frac {x^{3}}{\log {\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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