Optimal. Leaf size=24 \[ \frac {2 \text {Ei}(2 \log (c x))}{c^2}-\frac {x^2}{\log (c x)} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2306, 2309, 2178} \[ \frac {2 \text {Ei}(2 \log (c x))}{c^2}-\frac {x^2}{\log (c x)} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2309
Rubi steps
\begin {align*} \int \frac {x}{\log ^2(c x)} \, dx &=-\frac {x^2}{\log (c x)}+2 \int \frac {x}{\log (c x)} \, dx\\ &=-\frac {x^2}{\log (c x)}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (c x)\right )}{c^2}\\ &=\frac {2 \text {Ei}(2 \log (c x))}{c^2}-\frac {x^2}{\log (c x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {2 \text {Ei}(2 \log (c x))}{c^2}-\frac {x^2}{\log (c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 1.38 \[ -\frac {c^{2} x^{2} - 2 \, \log \left (c x\right ) \operatorname {log\_integral}\left (c^{2} x^{2}\right )}{c^{2} \log \left (c x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 24, normalized size = 1.00 \[ -\frac {x^{2}}{\log \left (c x\right )} + \frac {2 \, {\rm Ei}\left (2 \, \log \left (c x\right )\right )}{c^{2}} \]
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 \[ -\frac {x^{2}}{\ln \left (c x \right )}-\frac {2 \Ei \left (1, -2 \ln \left (c x \right )\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 13, normalized size = 0.54 \[ \frac {2 \, \Gamma \left (-1, -2 \, \log \left (c x\right )\right )}{c^{2}} \]
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 \[ \int \frac {x}{{\ln \left (c\,x\right )}^2} \,d x \]
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 \[ - \frac {x^{2}}{\log {\left (c x \right )}} + 2 \int \frac {x}{\log {\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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