Optimal. Leaf size=39 \[ \frac {1}{2} c \text {Ei}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
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Rubi [A] time = 0.05, antiderivative size = 39, 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} \[ \frac {1}{2} c \text {Ei}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2309
Rubi steps
\begin {align*} \int \frac {1}{x^2 \log ^3(c x)} \, dx &=-\frac {1}{2 x \log ^2(c x)}-\frac {1}{2} \int \frac {1}{x^2 \log ^2(c x)} \, dx\\ &=-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)}+\frac {1}{2} \int \frac {1}{x^2 \log (c x)} \, dx\\ &=-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (c x)\right )\\ &=\frac {1}{2} c \text {Ei}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ \frac {1}{2} c \text {Ei}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 34, normalized size = 0.87 \[ \frac {c x \log \left (c x\right )^{2} \operatorname {log\_integral}\left (\frac {1}{c x}\right ) + \log \left (c x\right ) - 1}{2 \, x \log \left (c x\right )^{2}} \]
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 \[ \int \frac {1}{x^{2} \log \left (c x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 0.85 \[ -\frac {c \Ei \left (1, \ln \left (c x \right )\right )}{2}+\frac {1}{2 x \ln \left (c x \right )}-\frac {1}{2 x \ln \left (c x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 9, normalized size = 0.23 \[ -c \Gamma \left (-2, \log \left (c x\right )\right ) \]
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 \[ \int \frac {1}{x^2\,{\ln \left (c\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {1}{x^{2} \log {\left (c x \right )}}\, dx}{2} + \frac {\log {\left (c x \right )} - 1}{2 x \log {\left (c x \right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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