Integrand size = 22, antiderivative size = 597 \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \]
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Time = 0.61 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {211, 2520, 12, 266, 6820, 5048, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {e} \sqrt {g}+i \sqrt {-d} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d} x+\sqrt {e}\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}} \]
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Rule 12
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2520
Rule 4940
Rule 4966
Rule 5048
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+(2 e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+\frac {e}{x^2}\right ) x^3} \, dx \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(2 e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (d+\frac {e}{x^2}\right ) x^3} \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(2 e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x \left (e+d x^2\right )} \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(2 e p) \int \left (\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e x}-\frac {d x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e \left (e+d x^2\right )}\right ) \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(2 p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {(2 d p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e+d x^2} \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(i p) \int \frac {\log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {(i p) \int \frac {\log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {(2 d p) \int \left (-\frac {\sqrt {-d} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+\frac {\left (\sqrt {-d} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {\left (\sqrt {-d} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-2 \frac {p \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{f}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {f} \left (-i \sqrt {-d}+\frac {\sqrt {e} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {f} \left (i \sqrt {-d}+\frac {\sqrt {e} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}-2 \frac {(i p) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{\sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}-\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\left (i \sqrt {-d} \sqrt {f}+\sqrt {e} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.18 \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+2 p \log \left (\frac {\sqrt {g} x}{\sqrt {-f}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (-\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-\log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-2 p \log \left (\frac {f \sqrt {g} x}{(-f)^{3/2}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt {g} \left (\sqrt {e}-\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} \left (\sqrt {e}+\sqrt {-d} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}-\sqrt {e} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt {-d} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt {-d} \sqrt {-f}+\sqrt {e} \sqrt {g}}\right )-2 p \operatorname {PolyLog}\left (2,1+\frac {\sqrt {g} x}{\sqrt {-f}}\right )+2 p \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {g} x}{(-f)^{3/2}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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\[\int \frac {\ln \left (c \left (d +\frac {e}{x^{2}}\right )^{p}\right )}{g \,x^{2}+f}d x\]
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\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+\frac {e}{x^2}\right )}^p\right )}{g\,x^2+f} \,d x \]
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