Integrand size = 27, antiderivative size = 377 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n} \]
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Time = 0.40 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {2525, 269, 272, 45, 2463, 2460, 720, 31, 649, 211, 266, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} n \left (d^2 f+e^2 g\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {-f} x^n+\sqrt {g}\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (f x^{2 n}+g\right )}+\frac {e^2 g p \log \left (f x^{2 n}+g\right )}{4 f^2 n \left (d^2 f+e^2 g\right )}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 n \left (d^2 f+e^2 g\right )}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{\sqrt {-f} d+e \sqrt {g}}\right )}{2 f^2 n} \]
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Rule 31
Rule 45
Rule 211
Rule 266
Rule 269
Rule 272
Rule 649
Rule 720
Rule 2438
Rule 2440
Rule 2441
Rule 2460
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x^2}\right )^2 x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {g x \log \left (c (d+e x)^p\right )}{f \left (g+f x^2\right )^2}+\frac {x \log \left (c (d+e x)^p\right )}{f \left (g+f x^2\right )}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{g+f x^2} \, dx,x,x^n\right )}{f n}-\frac {g \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\left (g+f x^2\right )^2} \, dx,x,x^n\right )}{f n} \\ & = \frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\text {Subst}\left (\int \left (-\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}-\sqrt {-f} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}+\sqrt {-f} x\right )}\right ) \, dx,x,x^n\right )}{f n}-\frac {(e g p) \text {Subst}\left (\int \frac {1}{(d+e x) \left (g+f x^2\right )} \, dx,x,x^n\right )}{2 f^2 n} \\ & = \frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}-\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}-\sqrt {-f} x} \, dx,x,x^n\right )}{2 (-f)^{3/2} n}+\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}+\sqrt {-f} x} \, dx,x,x^n\right )}{2 (-f)^{3/2} n}-\frac {(e g p) \text {Subst}\left (\int \frac {d f-e f x}{g+f x^2} \, dx,x,x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}-\frac {\left (e^3 g p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n} \\ & = -\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}+\sqrt {-f} x\right )}{-d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f^2 n}-\frac {(d e g p) \text {Subst}\left (\int \frac {1}{g+f x^2} \, dx,x,x^n\right )}{2 f \left (d^2 f+e^2 g\right ) n}+\frac {\left (e^2 g p\right ) \text {Subst}\left (\int \frac {x}{g+f x^2} \, dx,x,x^n\right )}{2 f \left (d^2 f+e^2 g\right ) n} \\ & = -\frac {d e \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-f} x}{-d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f^2 n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-f} x}{d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f^2 n} \\ & = -\frac {d e \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n} \\ \end{align*}
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 59.38 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.49
method | result | size |
risch | \(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) g}{2 n \,f^{2} \left (g +f \,x^{2 n}\right )}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-f g}+d f}\right )}{2 n \,f^{2}}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-f g}-d f}\right )}{2 n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-f g}+d f}\right )}{2 n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-f g}-d f}\right )}{2 n \,f^{2}}-\frac {e^{2} g p \ln \left (d +e \,x^{n}\right )}{2 f^{2} \left (d^{2} f +e^{2} g \right ) n}+\frac {e^{2} g p \ln \left (g +f \,x^{2 n}\right )}{4 f^{2} \left (d^{2} f +e^{2} g \right ) n}-\frac {p e g d \arctan \left (\frac {x^{n} f}{\sqrt {f g}}\right )}{2 n f \left (d^{2} f +e^{2} g \right ) \sqrt {f g}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {g}{2 n \,f^{2} \left (g +f \,x^{2 n}\right )}\right )\) | \(561\) |
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} x} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+\frac {g}{x^{2\,n}}\right )}^2} \,d x \]
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