Integrand size = 18, antiderivative size = 80 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p^2 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{a} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2504, 2444, 2441, 2352} \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}+\frac {b p^2 \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{a} \]
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Rule 2352
Rule 2441
Rule 2444
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {(b p) \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{a} \\ & = \frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (b^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{a} \\ & = \frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p^2 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {b \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^2}+\frac {b p^2 \operatorname {PolyLog}\left (2,\frac {a+b x^2}{a}\right )}{a} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 481, normalized size of antiderivative = 6.01
method | result | size |
risch | \(-\frac {{\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2 x^{2}}+\frac {2 p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{a}-\frac {p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (b \,x^{2}+a \right )}{a}-\frac {2 p^{2} b \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a}-\frac {2 p^{2} b \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a}-\frac {2 p^{2} b \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a}-\frac {2 p^{2} b \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{a}+\frac {p^{2} b \ln \left (b \,x^{2}+a \right )^{2}}{2 a}+\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2 x^{2}}+p b \left (\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a}\right )\right )-\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{8 x^{2}}\) | \(481\) |
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{3}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\frac {1}{2} \, b^{2} p^{2} {\left (\frac {\log \left (b x^{2} + a\right )^{2}}{a b} - \frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )}}{a b}\right )} - b p {\left (\frac {\log \left (b x^{2} + a\right )}{a} - \frac {\log \left (x^{2}\right )}{a}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{2 \, x^{2}} \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^3} \,d x \]
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