Optimal. Leaf size=24 \[ \frac {4 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{\log (c x)} \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2306, 2309, 2178} \[ \frac {4 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{\log (c x)} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2309
Rubi steps
\begin {align*} \int \frac {x^3}{\log ^2(c x)} \, dx &=-\frac {x^4}{\log (c x)}+4 \int \frac {x^3}{\log (c x)} \, dx\\ &=-\frac {x^4}{\log (c x)}+\frac {4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (c x)\right )}{c^4}\\ &=\frac {4 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{\log (c x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ \frac {4 \text {Ei}(4 \log (c x))}{c^4}-\frac {x^4}{\log (c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 33, normalized size = 1.38 \[ -\frac {c^{4} x^{4} - 4 \, \log \left (c x\right ) \operatorname {log\_integral}\left (c^{4} x^{4}\right )}{c^{4} \log \left (c x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 24, normalized size = 1.00 \[ -\frac {x^{4}}{\log \left (c x\right )} + \frac {4 \, {\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, normalized size = 1.08 \[ -\frac {x^{4}}{\ln \left (c x \right )}-\frac {4 \Ei \left (1, -4 \ln \left (c x \right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 13, normalized size = 0.54 \[ \frac {4 \, \Gamma \left (-1, -4 \, \log \left (c x\right )\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^3}{{\ln \left (c\,x\right )}^2} \,d x \]
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 \[ - \frac {x^{4}}{\log {\left (c x \right )}} + 4 \int \frac {x^{3}}{\log {\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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