Optimal. Leaf size=36 \[ 2 c^2 \text {Ei}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]
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Rubi [A] time = 0.05, antiderivative size = 36, 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} \[ 2 c^2 \text {Ei}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2309
Rubi steps
\begin {align*} \int \frac {1}{x^3 \log ^3(c x)} \, dx &=-\frac {1}{2 x^2 \log ^2(c x)}-\int \frac {1}{x^3 \log ^2(c x)} \, dx\\ &=-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)}+2 \int \frac {1}{x^3 \log (c x)} \, dx\\ &=-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)}+\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (c x)\right )\\ &=2 c^2 \text {Ei}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.00 \[ 2 c^2 \text {Ei}(-2 \log (c x))-\frac {1}{2 x^2 \log ^2(c x)}+\frac {1}{x^2 \log (c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 41, normalized size = 1.14 \[ \frac {4 \, c^{2} x^{2} \log \left (c x\right )^{2} \operatorname {log\_integral}\left (\frac {1}{c^{2} x^{2}}\right ) + 2 \, \log \left (c x\right ) - 1}{2 \, x^{2} \log \left (c x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \log \left (c x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 1.00 \[ -2 c^{2} \Ei \left (1, 2 \ln \left (c x \right )\right )+\frac {1}{x^{2} \ln \left (c x \right )}-\frac {1}{2 x^{2} \ln \left (c x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 13, normalized size = 0.36 \[ -4 \, c^{2} \Gamma \left (-2, 2 \, \log \left (c x\right )\right ) \]
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 \[ \int \frac {1}{x^3\,{\ln \left (c\,x\right )}^3} \,d x \]
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 \[ 2 \int \frac {1}{x^{3} \log {\left (c x \right )}}\, dx + \frac {2 \log {\left (c x \right )} - 1}{2 x^{2} \log {\left (c x \right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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