Integrand size = 22, antiderivative size = 360 \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (e+d x)}{\left (i d \sqrt {f}+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (e+d x)}{\left (i d \sqrt {f}+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \]
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Time = 0.32 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497} \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (d x+e)}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (e \sqrt {g}+i d \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (e+d x)}{\left (i \sqrt {f} d+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}} \]
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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 4996
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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+(e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+\frac {e}{x}\right ) x^2} \, dx \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (d+\frac {e}{x}\right ) 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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(e p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (e+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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {(e p) \int \left (\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e x}-\frac {d \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e (e+d x)}\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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {p \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {(d p) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e+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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {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} (e+d x)}{\left (i d \sqrt {f}+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\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} (e+d x)}{\sqrt {f} \left (i d+\frac {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 {(i p) \int \frac {\log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{2 \sqrt {f} \sqrt {g}}-\frac {(i p) \int \frac {\log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{x} \, dx}{2 \sqrt {f} \sqrt {g}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {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} (e+d x)}{\left (i d \sqrt {f}+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 )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} (e+d x)}{\left (i d \sqrt {f}+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}-\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}\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {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} (e+d x)}{\left (i d \sqrt {f}+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 )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (\frac {i \sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} (e+d x)}{\left (i d \sqrt {f}+e \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+p \log \left (\frac {\sqrt {g} x}{\sqrt {-f}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} (e+d x)}{d \sqrt {-f}+e \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-\log \left (c \left (d+\frac {e}{x}\right )^p\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-p \log \left (\frac {f \sqrt {g} x}{(-f)^{3/2}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} (e+d x)}{d \sqrt {-f}-e \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )-p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {-f}+e \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {-f}+\sqrt {g} x\right )}{d \sqrt {-f}-e \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,1+\frac {\sqrt {g} x}{\sqrt {-f}}\right )+p \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {g} x}{(-f)^{3/2}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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\[\int \frac {\ln \left (c \left (d +\frac {e}{x}\right )^{p}\right )}{g \,x^{2}+f}d x\]
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\[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
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none
Time = 0.33 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.05 \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\frac {e p {\left (\frac {4 \, \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (d + \frac {e}{x}\right )}{e} - \frac {{\left (\pi - 2 \, \arctan \left (\frac {{\left (d^{2} x + d e\right )} \sqrt {f} \sqrt {g}}{d^{2} f + e^{2} g}, \frac {d e g x + e^{2} g}{d^{2} f + e^{2} g}\right )\right )} \log \left (g x^{2} + f\right ) - 4 \, \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) + 2 \, \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {d^{2} g x^{2} + 2 \, d e g x + e^{2} g}{d^{2} f + e^{2} g}\right ) + 2 i \, {\rm Li}_2\left (\frac {i \, \sqrt {g} x + \sqrt {f}}{\sqrt {f}}\right ) - 2 i \, {\rm Li}_2\left (-\frac {i \, \sqrt {g} x - \sqrt {f}}{\sqrt {f}}\right ) + 2 i \, {\rm Li}_2\left (\frac {d e g x + d^{2} f - {\left (i \, d^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{d^{2} f + 2 i \, d e \sqrt {f} \sqrt {g} - e^{2} g}\right ) - 2 i \, {\rm Li}_2\left (\frac {d e g x + d^{2} f + {\left (i \, d^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{d^{2} f - 2 i \, d e \sqrt {f} \sqrt {g} - e^{2} g}\right )}{e}\right )}}{4 \, \sqrt {f g}} - \frac {p \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (d + \frac {e}{x}\right )}{\sqrt {f g}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right )}{\sqrt {f g}} \]
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\[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+\frac {e}{x}\right )}^p\right )}{g\,x^2+f} \,d x \]
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