Integrand size = 27, antiderivative size = 419 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {g} x^n}{\sqrt {f}}\right )}{2 f^{3/2} \left (e^2 f+d^2 g\right ) n}-\frac {e^2 p \log \left (d+e x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 p \log \left (f+g x^{2 n}\right )}{4 f \left (e^2 f+d^2 g\right ) n}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f^2 n} \]
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Time = 0.39 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {2525, 272, 46, 2463, 2441, 2352, 2460, 720, 31, 649, 211, 266, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {g} x^n}{\sqrt {f}}\right )}{2 f^{3/2} n \left (d^2 g+e^2 f\right )}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {e^2 p \log \left (f+g x^{2 n}\right )}{4 f n \left (d^2 g+e^2 f\right )}-\frac {e^2 p \log \left (d+e x^n\right )}{2 f n \left (d^2 g+e^2 f\right )}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} \left (e x^n+d\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {g} \left (e x^n+d\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f^2 n} \]
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Rule 31
Rule 46
Rule 211
Rule 266
Rule 272
Rule 649
Rule 720
Rule 2352
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 )}{x \left (f+g x^2\right )^2} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{f^2 x}-\frac {g x \log \left (c (d+e x)^p\right )}{f \left (f+g x^2\right )^2}-\frac {g x \log \left (c (d+e x)^p\right )}{f^2 \left (f+g x^2\right )}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{f+g x^2} \, dx,x,x^n\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\left (f+g x^2\right )^2} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \left (-\frac {\log \left (c (d+e x)^p\right )}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\log \left (c (d+e x)^p\right )}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx,x,x^n\right )}{f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {1}{(d+e x) \left (f+g x^2\right )} \, dx,x,x^n\right )}{2 f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n}+\frac {\sqrt {g} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}-\sqrt {g} x} \, dx,x,x^n\right )}{2 f^2 n}-\frac {\sqrt {g} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}+\sqrt {g} x} \, dx,x,x^n\right )}{2 f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {d g-e g x}{f+g x^2} \, dx,x,x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}-\frac {\left (e^3 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n} \\ & = -\frac {e^2 p \log \left (d+e x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \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 {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \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}{f+g x^2} \, dx,x,x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}+\frac {\left (e^2 g p\right ) \text {Subst}\left (\int \frac {x}{f+g x^2} \, dx,x,x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n} \\ & = -\frac {d e \sqrt {g} p \tan ^{-1}\left (\frac {\sqrt {g} x^n}{\sqrt {f}}\right )}{2 f^{3/2} \left (e^2 f+d^2 g\right ) n}-\frac {e^2 p \log \left (d+e x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 p \log \left (f+g x^{2 n}\right )}{4 f \left (e^2 f+d^2 g\right ) n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \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 {g} x}{e \sqrt {-f}+d \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 {g} x^n}{\sqrt {f}}\right )}{2 f^{3/2} \left (e^2 f+d^2 g\right ) n}-\frac {e^2 p \log \left (d+e x^n\right )}{2 f \left (e^2 f+d^2 g\right ) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 f n \left (f+g x^{2 n}\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x^n\right )}{e \sqrt {-f}+d \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}+\sqrt {g} x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 p \log \left (f+g x^{2 n}\right )}{4 f \left (e^2 f+d^2 g\right ) n}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2 n}-\frac {p \text {Li}_2\left (\frac {\sqrt {g} \left (d+e x^n\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{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 = 11.00 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n f \left (f +g \,x^{2 n}\right )}-\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (f +g \,x^{2 n}\right )}{2 n \,f^{2}}-\frac {e^{2} p \ln \left (d +e \,x^{n}\right )}{2 f \left (d^{2} g +f \,e^{2}\right ) n}+\frac {e^{2} p \ln \left (f +g \,x^{2 n}\right )}{4 f \left (d^{2} g +f \,e^{2}\right ) n}-\frac {p e g d \arctan \left (\frac {x^{n} g}{\sqrt {f g}}\right )}{2 n f \left (d^{2} g +f \,e^{2}\right ) \sqrt {f g}}-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (f +g \,x^{2 n}\right )}{2 n \,f^{2}}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}-g \left (d +e \,x^{n}\right )+d g}{e \sqrt {-f g}+d g}\right )}{2 n \,f^{2}}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-f g}+g \left (d +e \,x^{n}\right )-d g}{e \sqrt {-f g}-d g}\right )}{2 n \,f^{2}}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}-g \left (d +e \,x^{n}\right )+d g}{e \sqrt {-f g}+d g}\right )}{2 n \,f^{2}}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-f g}+g \left (d +e \,x^{n}\right )-d g}{e \sqrt {-f g}-d g}\right )}{2 n \,f^{2}}+\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 (x^{n}\right )}{n \,f^{2}}+\frac {1}{2 n f \left (f +g \,x^{2 n}\right )}-\frac {\ln \left (f +g \,x^{2 n}\right )}{2 n \,f^{2}}\right )\) | \(635\) |
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{2 \, n} + f\right )}^{2} x} \,d x } \]
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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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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{2 \, n} + f\right )}^{2} x} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{2 \, n} + f\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{2 n}\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+g\,x^{2\,n}\right )}^2} \,d x \]
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