Optimal. Leaf size=17 \[ \frac {\text {Li}_2\left (1-c x^2\right )}{2 c} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2336, 2315} \[ \frac {\text {PolyLog}\left (2,1-c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2336
Rubi steps
\begin {align*} \int \frac {x \log \left (c x^2\right )}{1-c x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (c x)}{1-c x} \, dx,x,x^2\right )\\ &=\frac {\text {Li}_2\left (1-c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ \frac {\text {Li}_2\left (1-c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 14, normalized size = 0.82 \[ \frac {{\rm Li}_2\left (-c x^{2} + 1\right )}{2 \, c} \]
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 \[ \int -\frac {x \log \left (c x^{2}\right )}{c x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 0.71 \[ \frac {\dilog \left (c \,x^{2}\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 76, normalized size = 4.47 \[ -\frac {\log \left (c x^{2} - 1\right ) \log \left (c x^{2}\right )}{2 \, c} + \frac {\log \left (c x^{2} - 1\right ) \log \relax (x)}{c} + \frac {\log \left (c x^{2} - 1\right ) \log \left (c x^{2}\right ) - 2 \, \log \left (c x^{2} - 1\right ) \log \relax (x) + {\rm Li}_2\left (-c x^{2} + 1\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 11, normalized size = 0.65 \[ \frac {{\mathrm {Li}}_{\mathrm {2}}\left (c\,x^2\right )}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.25, size = 78, normalized size = 4.59 \[ \frac {\begin {cases} i \pi \log {\relax (x )} - \frac {\operatorname {Li}_{2}\left (c x^{2}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (c x^{2}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \frac {\operatorname {Li}_{2}\left (c x^{2}\right )}{2} & \text {otherwise} \end {cases}}{c} - \frac {\log {\left (c x^{2} \right )} \log {\left (c x^{2} - 1 \right )}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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