Optimal. Leaf size=45 \[ -\frac {\log ^3(c x)}{2 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3}{8 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2305, 2304} \[ -\frac {\log ^3(c x)}{2 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int \frac {\log ^3(c x)}{x^3} \, dx &=-\frac {\log ^3(c x)}{2 x^2}+\frac {3}{2} \int \frac {\log ^2(c x)}{x^3} \, dx\\ &=-\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2}+\frac {3}{2} \int \frac {\log (c x)}{x^3} \, dx\\ &=-\frac {3}{8 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {\log ^3(c x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 45, normalized size = 1.00 \[ -\frac {\log ^3(c x)}{2 x^2}-\frac {3 \log ^2(c x)}{4 x^2}-\frac {3 \log (c x)}{4 x^2}-\frac {3}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 0.64 \[ -\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 37, normalized size = 0.82 \[ -\frac {\log \left (c x\right )^{3}}{2 \, x^{2}} - \frac {3 \, \log \left (c x\right )^{2}}{4 \, x^{2}} - \frac {3 \, \log \left (c x\right )}{4 \, x^{2}} - \frac {3}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 0.84 \[ -\frac {\ln \left (c x \right )^{3}}{2 x^{2}}-\frac {3 \ln \left (c x \right )^{2}}{4 x^{2}}-\frac {3 \ln \left (c x \right )}{4 x^{2}}-\frac {3}{8 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 29, normalized size = 0.64 \[ -\frac {4 \, \log \left (c x\right )^{3} + 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 3}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 29, normalized size = 0.64 \[ -\frac {\frac {{\ln \left (c\,x\right )}^3}{2}+\frac {3\,{\ln \left (c\,x\right )}^2}{4}+\frac {3\,\ln \left (c\,x\right )}{4}+\frac {3}{8}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 44, normalized size = 0.98 \[ - \frac {\log {\left (c x \right )}^{3}}{2 x^{2}} - \frac {3 \log {\left (c x \right )}^{2}}{4 x^{2}} - \frac {3 \log {\left (c x \right )}}{4 x^{2}} - \frac {3}{8 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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