Optimal. Leaf size=257 \[ -\frac {e g^2 p x^{-3 n}}{12 d n}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}-\frac {e f g p x^{-n}}{d n}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}-\frac {e^2 f g p \log (x)}{d^2}-\frac {e^4 g^2 p \log (x)}{4 d^4}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.20, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2525, 269,
272, 45, 2463, 2442, 46, 2441, 2352} \begin {gather*} \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {e^4 g^2 p \log (x)}{4 d^4}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}-\frac {e^2 f g p \log (x)}{d^2}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}-\frac {e f g p x^{-n}}{d n}-\frac {e g^2 p x^{-3 n}}{12 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 46
Rule 269
Rule 272
Rule 2352
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^{-2 n}\right )^2 \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 )^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {g^2 \log \left (c (d+e x)^p\right )}{x^5}+\frac {2 f g \log \left (c (d+e x)^p\right )}{x^3}+\frac {f^2 \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {(2 f g) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^n\right )}{n}+\frac {g^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^5} \, dx,x,x^n\right )}{n}\\ &=-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e f g p) \text {Subst}\left (\int \frac {1}{x^2 (d+e x)} \, dx,x,x^n\right )}{n}+\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \frac {1}{x^4 (d+e x)} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}+\frac {(e f 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 )}{n}+\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \left (\frac {1}{d x^4}-\frac {e}{d^2 x^3}+\frac {e^2}{d^3 x^2}-\frac {e^3}{d^4 x}+\frac {e^4}{d^4 (d+e x)}\right ) \, dx,x,x^n\right )}{4 n}\\ &=-\frac {e g^2 p x^{-3 n}}{12 d n}+\frac {e^2 g^2 p x^{-2 n}}{8 d^2 n}-\frac {e f g p x^{-n}}{d n}-\frac {e^3 g^2 p x^{-n}}{4 d^3 n}-\frac {e^2 f g p \log (x)}{d^2}-\frac {e^4 g^2 p \log (x)}{4 d^4}+\frac {e^2 f g p \log \left (d+e x^n\right )}{d^2 n}+\frac {e^4 g^2 p \log \left (d+e x^n\right )}{4 d^4 n}-\frac {g^2 x^{-4 n} \log \left (c \left (d+e x^n\right )^p\right )}{4 n}-\frac {f g x^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 204, normalized size = 0.79 \begin {gather*} \frac {g \left (6 e^2 \left (4 d^2 f+e^2 g\right ) p \log \left (e-e x^{-n}\right )-d x^{-4 n} \left (e p x^n \left (-3 d e g x^n+6 e^2 g x^{2 n}+2 d^2 \left (g+12 f x^{2 n}\right )\right )+6 d^3 \left (g+4 f x^{2 n}\right ) \log \left (c \left (d+e x^n\right )^p\right )\right )\right )-6 n \log (x) \left (e^2 g \left (4 d^2 f+e^2 g\right ) p-4 d^4 f^2 \log \left (c \left (d+e x^n\right )^p\right )+4 d^4 f^2 p \log \left (1+\frac {e x^n}{d}\right )\right )-24 d^4 f^2 p \text {Li}_2\left (-\frac {e x^n}{d}\right )}{24 d^4 n} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.46, size = 755, normalized size = 2.94
method | result | size |
risch | \(\frac {e^{2} f g p \ln \left (d +e \,x^{n}\right )}{d^{2} n}+\frac {e^{4} g^{2} p \ln \left (d +e \,x^{n}\right )}{4 d^{4} n}-\frac {p \,f^{2} \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )+\frac {\ln \left (c \right ) f^{2} \ln \left (x^{n}\right )}{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^{2} \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} f^{2} \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^{2} \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f^{2} \ln \left (x^{n}\right )}{2 n}-\frac {\ln \left (c \right ) f g \,x^{-2 n}}{n}-\frac {e f g p \,x^{-n}}{d 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^{2} x^{-4 n}}{8 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f g \,x^{-2 n}}{2 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^{2} x^{-4 n}}{8 n}-\frac {p \,e^{2} f g \ln \left (x^{n}\right )}{n \,d^{2}}+\frac {\left (4 f^{2} \ln \left (x \right ) n \,x^{4 n}-4 f g \,x^{2 n}-g^{2}\right ) x^{-4 n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{4 n}-\frac {\ln \left (c \right ) g^{2} x^{-4 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 g \,x^{-2 n}}{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} f g \,x^{-2 n}}{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^{2} x^{-4 n}}{8 n}-\frac {e^{3} g^{2} p \,x^{-n}}{4 d^{3} n}-\frac {p e \,g^{2} x^{-3 n}}{12 n d}+\frac {p \,e^{2} g^{2} x^{-2 n}}{8 n \,d^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g^{2} x^{-4 n}}{8 n}-\frac {p \,e^{4} g^{2} \ln \left (x^{n}\right )}{4 n \,d^{4}}+\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 g \,x^{-2 n}}{2 n}\) | \(755\) |
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 = 0.40, size = 265, normalized size = 1.03 \begin {gather*} -\frac {24 \, d^{4} f^{2} n p x^{4 \, n} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + 24 \, d^{4} f^{2} p x^{4 \, n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) + 2 \, d^{3} g^{2} p x^{n} e + 6 \, d^{4} g^{2} \log \left (c\right ) - 6 \, {\left (4 \, d^{4} f^{2} n \log \left (c\right ) - 4 \, d^{2} f g n p e^{2} - g^{2} n p e^{4}\right )} x^{4 \, n} \log \left (x\right ) + 6 \, {\left (4 \, d^{3} f g p e + d g^{2} p e^{3}\right )} x^{3 \, n} + 3 \, {\left (8 \, d^{4} f g \log \left (c\right ) - d^{2} g^{2} p e^{2}\right )} x^{2 \, n} + 6 \, {\left (4 \, d^{4} f g p x^{2 \, n} + d^{4} g^{2} p - {\left (4 \, d^{4} f^{2} n p \log \left (x\right ) + 4 \, d^{2} f g p e^{2} + g^{2} p e^{4}\right )} x^{4 \, n}\right )} \log \left (x^{n} e + d\right )}{24 \, d^{4} n x^{4 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+\frac {g}{x^{2\,n}}\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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