Integrand size = 18, antiderivative size = 72 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )-p^2 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2504, 2443, 2481, 2421, 6724} \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p^2 \left (-\operatorname {PolyLog}\left (3,\frac {b x^2}{a}+1\right )\right ) \]
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Rule 2421
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-(b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-p \text {Subst}\left (\int \frac {\log \left (c x^p\right ) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x^2\right ) \\ & = \frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-p^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right ) \\ & = \frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+p \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-p^2 \text {Li}_3\left (1+\frac {b x^2}{a}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(72)=144\).
Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.26 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\log (x) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+2 p \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (\log (x) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {b x^2}{a}\right )\right )+\frac {1}{2} p^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )+2 \log \left (a+b x^2\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right )\right ) \]
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\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{x}d x\]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.64 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\frac {1}{2} \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} p^{2} + {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} p \log \left (c\right ) + \log \left (c\right )^{2} \log \left (x\right ) \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x} \,d x \]
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