Integrand size = 25, antiderivative size = 204 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=-\frac {e p \log \left (d+e x^n\right )}{f (e f-d g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {e p \log \left (f+g x^n\right )}{f (e f-d g) n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f^2 n} \]
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Time = 0.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2525, 46, 2463, 2441, 2352, 2442, 36, 31, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^n+d\right )}{e f-d g}\right )}{f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f^2 n}-\frac {e p \log \left (d+e x^n\right )}{f n (e f-d g)}+\frac {e p \log \left (f+g x^n\right )}{f n (e f-d g)} \]
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Rule 31
Rule 36
Rule 46
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
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 (f+g x)^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 \log \left (c (d+e x)^p\right )}{f (f+g x)^2}-\frac {g \log \left (c (d+e x)^p\right )}{f^2 (f+g x)}\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 {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^n\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^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 (f+g x^n\right )}{e f-d g}\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 {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^n\right )}{f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^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 (f+g x^n\right )}{e f-d g}\right )}{f^2 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 {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^n\right )}{f^2 n}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{f (e f-d g) n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^n\right )}{f (e f-d g) n} \\ & = -\frac {e p \log \left (d+e x^n\right )}{f (e f-d g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^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 {e p \log \left (f+g x^n\right )}{f (e f-d g) n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}-\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\frac {-\frac {e f p \log \left (d+e x^n\right )}{e f-d g}+\frac {f \log \left (c \left (d+e x^n\right )^p\right )}{f+g x^n}+\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {e f p \log \left (f+g x^n\right )}{e f-d g}-\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )-p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^n\right )}{-e f+d g}\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f^2 n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.93 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.03
method | result | size |
risch | \(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (x^{n}\right )}{n \,f^{2}}-\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (f +g \,x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n f \left (f +g \,x^{n}\right )}+\frac {p e \ln \left (d +e \,x^{n}\right )}{n f \left (d g -e f \right )}-\frac {p e \ln \left (f +g \,x^{n}\right )}{n f \left (d g -e f \right )}-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}+\frac {p \ln \left (f +g \,x^{n}\right ) \ln \left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (x^{n}\right )}{f^{2}}-\frac {\ln \left (f +g \,x^{n}\right )}{f^{2}}+\frac {1}{f \left (f +g \,x^{n}\right )}\right )}{n}\) | \(414\) |
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{n} + f\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.14 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=-e n p {\left (\frac {\log \left (\frac {e x^{n} + d}{e}\right )}{e f^{2} n^{2} - d f g n^{2}} - \frac {\log \left (\frac {g x^{n} + f}{g}\right )}{e f^{2} n^{2} - d f g n^{2}} + \frac {\log \left (x^{n}\right ) \log \left (\frac {e x^{n}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e x^{n}}{d}\right )}{e f^{2} n^{2}} - \frac {\log \left (g x^{n} + f\right ) \log \left (-\frac {e g x^{n} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{n} + e f}{e f - d g}\right )}{e f^{2} n^{2}}\right )} + {\left (\frac {1}{f g n x^{n} + f^{2} n} - \frac {\log \left (g x^{n} + f\right )}{f^{2} n} + \frac {\log \left (x^{n}\right )}{f^{2} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{n} + f\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+g\,x^n\right )}^2} \,d x \]
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