Optimal. Leaf size=126 \[ -\frac {e g p x^{-n}}{2 d n}-\frac {e^2 g p \log (x)}{2 d^2}+\frac {e^2 g p \log \left (d+e x^n\right )}{2 d^2 n}-\frac {g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2525, 14, 2463,
2442, 46, 2441, 2352} \begin {gather*} \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\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 {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 {e g p x^{-n}}{2 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 46
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{-2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\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*}
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Mathematica [A]
time = 0.30, size = 148, normalized size = 1.17 \begin {gather*} -\frac {d e g p x^{-n}-2 d^2 f p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right )-e^2 g p \log \left (n \left (d+e x^n\right )\right )+d^2 g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )+n \log (x) \left (e^2 g p+2 d^2 f p \log \left (d+e x^n\right )-2 d^2 f \log \left (c \left (d+e x^n\right )^p\right )\right )-2 d^2 f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{2 d^2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 448, normalized size = 3.56
method | result | size |
risch | \(\frac {\left (2 f \ln \left (x \right ) n \,x^{2 n}-g \right ) x^{-2 n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{2 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) g \,x^{-2 n}}{4 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} g \,x^{-2 n}}{4 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) g \,x^{-2 n}}{4 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g \,x^{-2 n}}{4 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}+\frac {\ln \left (c \right ) f \ln \left (x^{n}\right )}{n}-\frac {\ln \left (c \right ) g \,x^{-2 n}}{2 n}+\frac {e^{2} g p \ln \left (d +e \,x^{n}\right )}{2 d^{2} n}-\frac {e g p \,x^{-n}}{2 d n}-\frac {p \,e^{2} g \ln \left (x^{n}\right )}{2 n \,d^{2}}-\frac {p f \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(448\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 154, normalized size = 1.22 \begin {gather*} -\frac {2 \, d^{2} f n p x^{2 \, n} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + 2 \, d^{2} f p x^{2 \, n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) + d g p x^{n} e + d^{2} g \log \left (c\right ) - {\left (2 \, d^{2} f n \log \left (c\right ) - g n p e^{2}\right )} x^{2 \, n} \log \left (x\right ) + {\left (d^{2} g p - {\left (2 \, d^{2} f n p \log \left (x\right ) + g p e^{2}\right )} x^{2 \, n}\right )} \log \left (x^{n} e + d\right )}{2 \, d^{2} n x^{2 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^{2\,n}}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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