Integrand size = 25, antiderivative size = 126 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {e g p x^{-n}}{2 d n}-\frac {e^2 g p \log (x)}{2 d^2}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2 n}-\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
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Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2525, 14, 2463, 2442, 46, 2441, 2352} \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2 n}-\frac {e^2 g p \log (x)}{2 d^2}+\frac {f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}-\frac {e g p x^{-n}}{2 d n} \]
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Rule 14
Rule 46
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (f+\frac {g}{x^2}\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {g \log \left (c (d+e x)^p\right )}{x^3}+\frac {f \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^n\right )}{n} \\ & = -\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(e f p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,x^n\right )}{2 n} \\ & = -\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \left (\frac {1}{d x^2}-\frac {e}{d^2 x}+\frac {e^2}{d^2 (d+e x)}\right ) \, dx,x,x^n\right )}{2 n} \\ & = -\frac {e g p x^{-n}}{2 d n}-\frac {e^2 g p \log (x)}{2 d^2}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2 n}-\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {\frac {e g p x^{-n} \left (d+e n x^n \log (x)-e x^n \log \left (d+e x^n\right )\right )}{d^2}+g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )-2 f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )\right )}{2 n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.64 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.12
method | result | size |
risch | \(\frac {\left (2 f \ln \left (x \right ) n \,x^{2 n}-g \right ) x^{-2 n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {f \ln \left (x^{n}\right )}{n}-\frac {g \,x^{-2 n}}{2 n}\right )+\frac {e^{2} g p \ln \left (d +e \,x^{n}\right )}{2 d^{2} n}-\frac {e g p \,x^{-n}}{2 d n}-\frac {p \,e^{2} g \ln \left (x^{n}\right )}{2 n \,d^{2}}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(267\) |
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Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.19 \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {2 \, d^{2} f n p x^{2 \, n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + 2 \, d^{2} f p x^{2 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d e g p x^{n} + d^{2} g \log \left (c\right ) + {\left (e^{2} g n p - 2 \, d^{2} f n \log \left (c\right )\right )} x^{2 \, n} \log \left (x\right ) + {\left (d^{2} g p - {\left (2 \, d^{2} f n p \log \left (x\right ) + e^{2} g p\right )} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \]
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\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {x^{- 2 n} \left (f x^{2 n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^{2\,n}}\right )}{x} \,d x \]
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