Optimal. Leaf size=156 \[ \frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (f x^n+g\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f^2 n}+\frac {e g p \log \left (d+e x^n\right )}{f^2 n (d f-e g)}-\frac {e g p \log \left (f x^n+g\right )}{f^2 n (d f-e g)} \]
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Rubi [A] time = 0.28, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2475, 263, 43, 2416, 2395, 36, 31, 2394, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f^2 n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (f x^n+g\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f^2 n}+\frac {e g p \log \left (d+e x^n\right )}{f^2 n (d f-e g)}-\frac {e g p \log \left (f x^n+g\right )}{f^2 n (d f-e g)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 43
Rule 263
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x}\right )^2 x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {g \log \left (c (d+e x)^p\right )}{f (g+f x)^2}+\frac {\log \left (c (d+e x)^p\right )}{f (g+f x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{g+f x} \, dx,x,x^n\right )}{f n}-\frac {g \operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(g+f x)^2} \, dx,x,x^n\right )}{f n}\\ &=\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (g+f x^n\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f^2 n}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx,x,x^n\right )}{f^2 n}-\frac {(e g p) \operatorname {Subst}\left (\int \frac {1}{(d+e x) (g+f x)} \, dx,x,x^n\right )}{f^2 n}\\ &=\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (g+f x^n\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f^2 n}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f x}{-d f+e g}\right )}{x} \, dx,x,d+e x^n\right )}{f^2 n}+\frac {\left (e^2 g p\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{f^2 (d f-e g) n}-\frac {(e g p) \operatorname {Subst}\left (\int \frac {1}{g+f x} \, dx,x,x^n\right )}{f (d f-e g) n}\\ &=\frac {e g p \log \left (d+e x^n\right )}{f^2 (d f-e g) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (g+f x^n\right )}-\frac {e g p \log \left (g+f x^n\right )}{f^2 (d f-e g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f^2 n}\\ \end {align*}
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Mathematica [B] time = 1.51, size = 433, normalized size = 2.78 \[ \frac {g \log \left (f-f x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )-f x^n \log \left (c \left (d+e x^n\right )^p\right )+f x^n \log \left (f-f x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \log \left (d x^{-n}+e\right ) \left (\left (f x^n+g\right ) \log \left (f-f x^{-n}\right )-f x^n\right )+p \left (f x^n+g\right ) \text {Li}_2\left (-\frac {f x^n}{g}\right )+g p \log \left (f-f x^{-n}\right )-g n p \log (x) \log \left (f-f x^{-n}\right )+f n p x^n \log (x) \log \left (\frac {f x^n}{g}+1\right )+g n p \log (x) \log \left (\frac {f x^n}{g}+1\right )+f p x^n \log \left (f-f x^{-n}\right )-f n p x^n \log (x) \log \left (f-f x^{-n}\right )}{f^2 n \left (f x^n+g\right )}-\frac {p \left (-\text {Li}_2\left (-\frac {g \left (d x^{-n}+e\right )}{d f-e g}\right )-\frac {d f \log \left (d x^{-n}+e\right )}{d f-e g}+\frac {d f \log \left (f+g x^{-n}\right )}{d f-e g}-\log \left (d x^{-n}+e\right ) \log \left (\frac {d \left (f+g x^{-n}\right )}{d f-e g}\right )+\frac {f x^n \log \left (d x^{-n}+e\right )}{f x^n+g}+\text {Li}_2\left (\frac {d x^{-n}}{e}+1\right )+\log \left (-\frac {d x^{-n}}{e}\right ) \log \left (d x^{-n}+e\right )\right )}{f^2 n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f^{2} x + \frac {2 \, f g x x^{n}}{x^{2 \, n}} + \frac {g^{2} x}{x^{2 \, n}}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.62, size = 589, normalized size = 3.78 \[ -\frac {e g p \ln \left (f \,x^{n}+g \right )}{\left (d f -e g \right ) f^{2} n}+\frac {e g p \ln \left (d f -e g +\left (f \,x^{n}+g \right ) e \right )}{\left (d f -e g \right ) f^{2} n}-\frac {i \pi g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )}{2 \left (f \,x^{n}+g \right ) f^{2} n}+\frac {i \pi g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 \left (f \,x^{n}+g \right ) f^{2} n}+\frac {i \pi g \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 \left (f \,x^{n}+g \right ) f^{2} n}-\frac {i \pi g \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{2 \left (f \,x^{n}+g \right ) f^{2} n}-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{n}+g \right )}{2 f^{2} n}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{n}+g \right )}{2 f^{2} n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{n}+g \right )}{2 f^{2} n}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (f \,x^{n}+g \right )}{2 f^{2} n}-\frac {p \ln \left (\frac {d f -e g +\left (f \,x^{n}+g \right ) e}{d f -e g}\right ) \ln \left (f \,x^{n}+g \right )}{f^{2} n}+\frac {g \ln \relax (c )}{\left (f \,x^{n}+g \right ) f^{2} n}+\frac {g \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{\left (f \,x^{n}+g \right ) f^{2} n}-\frac {p \dilog \left (\frac {d f -e g +\left (f \,x^{n}+g \right ) e}{d f -e g}\right )}{f^{2} n}+\frac {\ln \relax (c ) \ln \left (f \,x^{n}+g \right )}{f^{2} n}+\frac {\ln \left (\left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{n}+g \right )}{f^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 209, normalized size = 1.34 \[ e n p {\left (\frac {d \log \left (\frac {e x^{n} + d}{e}\right )}{d e f^{2} n^{2} - e^{2} f g n^{2}} - \frac {g \log \left (\frac {f x^{n} + g}{f}\right )}{d f^{3} n^{2} - e f^{2} g n^{2}} - \frac {\log \left (f x^{n} + g\right ) \log \left (\frac {e f x^{n} + e g}{d f - e g} + 1\right ) + {\rm Li}_2\left (-\frac {e f x^{n} + e g}{d f - e g}\right )}{e f^{2} n^{2}}\right )} - {\left (\frac {1}{f^{2} n + \frac {f g n}{x^{n}}} - \frac {\log \left (f + \frac {g}{x^{n}}\right )}{f^{2} n} + \frac {\log \left (\frac {1}{x^{n}}\right )}{f^{2} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+\frac {g}{x^n}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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