Integrand size = 18, antiderivative size = 119 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{a}-\frac {3 b p^3 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right )}{a} \]
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Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2444, 2443, 2481, 2421, 6724} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {3 b p^2 \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {3 b p^3 \operatorname {PolyLog}\left (3,\frac {b x^2}{a}+1\right )}{a} \]
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Rule 2421
Rule 2443
Rule 2444
Rule 2481
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^3\left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {(3 b p) \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 a} \\ & = \frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (3 b^2 p^2\right ) \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 )}{a} \\ & = \frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (3 b p^2\right ) \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 )}{a} \\ & = \frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}-\frac {\left (3 b p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a} \\ & = \frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}-\frac {3 b p^3 \text {Li}_3\left (1+\frac {b x^2}{a}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(119)=238\).
Time = 0.24 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.54 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=-\frac {-6 b p^3 x^2 \log (x) \log ^2\left (a+b x^2\right )+3 b p^3 x^2 \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )+b p^3 x^2 \log ^3\left (a+b x^2\right )+12 b p^2 x^2 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b p^2 x^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-3 b p^2 x^2 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b p x^2 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+3 b p x^2 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+a \log ^3\left (c \left (a+b x^2\right )^p\right )-6 b p^2 x^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )+6 b p^3 x^2 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right )}{2 a x^2} \]
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\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{3}}d x\]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.70 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {1}{2} \, {\left (\frac {3 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} p^{2}}{a} + \frac {6 \, {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} p \log \left (c\right )}{a} + \frac {6 \, \log \left (c\right )^{2} \log \left (x\right )}{a} - \frac {p^{2} \log \left (b x^{2} + a\right )^{3} + 3 \, p \log \left (b x^{2} + a\right )^{2} \log \left (c\right ) + 3 \, \log \left (b x^{2} + a\right ) \log \left (c\right )^{2}}{a}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{2 \, x^{2}} \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^3} \,d x \]
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