Integrand size = 18, antiderivative size = 190 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {4 i \sqrt {b} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {8 \sqrt {b} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}}+\frac {4 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+\frac {4 i \sqrt {b} p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}} \]
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Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2507, 211, 2520, 12, 5040, 4964, 2449, 2352} \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {4 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}+\frac {4 i \sqrt {b} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {8 \sqrt {b} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+\frac {4 i \sqrt {b} p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{\sqrt {a}} \]
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Rule 12
Rule 211
Rule 2352
Rule 2449
Rule 2507
Rule 2520
Rule 4964
Rule 5040
Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+(4 b p) \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx \\ & = \frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}-\left (8 b^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a+b x^2\right )} \, dx \\ & = \frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}-\frac {\left (8 b^{3/2} p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a+b x^2} \, dx}{\sqrt {a}} \\ & = \frac {4 i \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+\frac {\left (8 b p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{i-\frac {\sqrt {b} x}{\sqrt {a}}} \, dx}{a} \\ & = \frac {4 i \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {8 \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}}+\frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}-\frac {\left (8 b p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{1+\frac {b x^2}{a}} \, dx}{a} \\ & = \frac {4 i \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {8 \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}}+\frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+\frac {\left (8 i \sqrt {b} p^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {b} x}{\sqrt {a}}}\right )}{\sqrt {a}} \\ & = \frac {4 i \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a}}+\frac {8 \sqrt {b} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}}+\frac {4 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {a}}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x}+\frac {4 i \sqrt {b} p^2 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {a}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.91 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {4 i \sqrt {b} p^2 x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2-\sqrt {a} \log ^2\left (c \left (a+b x^2\right )^p\right )+4 \sqrt {b} p x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (2 p \log \left (\frac {2 i}{i-\frac {\sqrt {b} x}{\sqrt {a}}}\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )+4 i \sqrt {b} p^2 x \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )}{\sqrt {a} x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.49 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.35
method | result | size |
risch | \(-\frac {{\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}^{2}}{x}-\frac {4 p^{2} b \arctan \left (\frac {b x}{\sqrt {a b}}\right ) \ln \left (b \,x^{2}+a \right )}{\sqrt {a b}}+\frac {4 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right ) \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{\sqrt {a b}}+p^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (b \,x^{2}+a \right )-b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )+\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 )}{x}+\frac {2 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\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}}{4 x}\) | \(446\) |
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{2}}\, dx \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^2} \,d x \]
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