Integrand size = 25, antiderivative size = 144 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {d^2 g p x^n}{3 e^2 n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
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Time = 0.11 (sec) , antiderivative size = 144, 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, 2441, 2352, 2442, 45} \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^2 g p x^n}{3 e^2 n}+\frac {f p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n} \]
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Rule 14
Rule 45
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (f+g x^3\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {f \log \left (c (d+e x)^p\right )}{x}+g x^2 \log \left (c (d+e x)^p\right )\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 x^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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 {x^3}{d+e x} \, dx,x,x^n\right )}{3 n} \\ & = \frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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 {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,x^n\right )}{3 n} \\ & = -\frac {d^2 g p x^n}{3 e^2 n}+\frac {d g p x^{2 n}}{6 e n}-\frac {g p x^{3 n}}{9 n}+\frac {d^3 g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 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 = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.82 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-\frac {g p \left (e x^n \left (6 d^2-3 d e x^n+2 e^2 x^{2 n}\right )-6 d^3 \log \left (d+e x^n\right )\right )}{e^3}+6 g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )+18 f \left (\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )\right )}{18 n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.65 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.85
method | result | size |
risch | \(\frac {\left (g \,x^{3 n}+3 f \ln \left (x \right ) n \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{3 n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (f \ln \left (x \right )+\frac {g \,x^{3 n}}{3 n}\right )-\frac {g p \,x^{3 n}}{9 n}+\frac {d g p \,x^{2 n}}{6 e n}-\frac {d^{2} g p \,x^{n}}{3 e^{2} n}+\frac {d^{3} g p \ln \left (d +e \,x^{n}\right )}{3 e^{3} n}-\frac {p f \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(267\) |
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Time = 0.42 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.03 \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=-\frac {18 \, e^{3} f n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 18 \, e^{3} f n \log \left (c\right ) \log \left (x\right ) - 3 \, d e^{2} g p x^{2 \, n} + 6 \, d^{2} e g p x^{n} + 18 \, e^{3} f p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + 2 \, {\left (e^{3} g p - 3 \, e^{3} g \log \left (c\right )\right )} x^{3 \, n} - 6 \, {\left (3 \, e^{3} f n p \log \left (x\right ) + e^{3} g p x^{3 \, n} + d^{3} g p\right )} \log \left (e x^{n} + d\right )}{18 \, e^{3} n} \]
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\[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f + g x^{3 n}\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \]
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\[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{3 \, n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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\[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{3 \, n} + f\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (f+g x^{3 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x^{3\,n}\right )}{x} \,d x \]
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