Integrand size = 25, antiderivative size = 97 \[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
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Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2525, 14, 2463, 2442, 36, 29, 31, 2441, 2352} \[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}-\frac {e g p \log \left (d+e x^n\right )}{d n}+\frac {e g p \log (x)}{d} \]
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Rule 14
Rule 29
Rule 31
Rule 36
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}\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^2}+\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^2} \, dx,x,x^n\right )}{n} \\ & = -\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{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 (d+e x)} \, dx,x,x^n\right )}{n} \\ & = -\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{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 \frac {1}{x} \, dx,x,x^n\right )}{d n}-\frac {\left (e^2 g p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d n} \\ & = \frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{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 = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {e g n p \log (x)-e g p \log \left (d+e x^n\right )-d g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )+d f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+d f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{d n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.90 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.49
method | result | size |
risch | \(\frac {\left (f \ln \left (x \right ) n \,x^{n}-g \right ) x^{-n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {g \,x^{-n}}{n}+\frac {f \ln \left (x^{n}\right )}{n}\right )-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )-\frac {e g p \ln \left (d +e \,x^{n}\right )}{d n}+\frac {p e g \ln \left (x^{n}\right )}{n d}\) | \(242\) |
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Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {d f n p x^{n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + d f p x^{n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d g \log \left (c\right ) - {\left (e g n p + d f n \log \left (c\right )\right )} x^{n} \log \left (x\right ) + {\left (d g p - {\left (d f n p \log \left (x\right ) - e g p\right )} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
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\[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {x^{- n} \left (f x^{n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{n}}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^n}\right )}{x} \,d x \]
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