Optimal. Leaf size=204 \[ -\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}-\frac {p \text {Li}_2\left (-\frac {g \left (e x^n+d\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{f^2 n}-\frac {e p \log \left (d+e x^n\right )}{f n (e f-d g)}+\frac {e p \log \left (f+g x^n\right )}{f n (e f-d g)} \]
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Rubi [A] time = 0.27, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2475, 44, 2416, 2394, 2315, 2395, 36, 31, 2393, 2391} \[ -\frac {p \text {PolyLog}\left (2,-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}-\frac {e p \log \left (d+e x^n\right )}{f n (e f-d g)}+\frac {e p \log \left (f+g x^n\right )}{f n (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 44
Rule 2315
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 )}{x (f+g x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {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 {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f^2 n}-\frac {g \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{f (e f-d g) n}+\frac {(e g p) \operatorname {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*}
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Mathematica [A] time = 0.16, size = 171, normalized size = 0.84 \[ \frac {\frac {f \log \left (c \left (d+e x^n\right )^p\right )}{f+g x^n}-\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )+\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \text {Li}_2\left (\frac {g \left (e x^n+d\right )}{d g-e f}\right )-\frac {e f p \log \left (d+e x^n\right )}{e f-d g}+\frac {e f p \log \left (f+g x^n\right )}{e f-d g}+p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{f^2 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{g^{2} x x^{2 \, n} + 2 \, f g x x^{n} + f^{2} x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{n} + f\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.71, size = 805, normalized size = 3.95 \[ \frac {e p \ln \left (e \,x^{n}+d \right )}{\left (d g -e f \right ) f n}-\frac {e p \ln \left (g \,x^{n}+f \right )}{\left (d g -e f \right ) f 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 )}{2 \left (g \,x^{n}+f \right ) f n}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2}}{2 \left (g \,x^{n}+f \right ) f 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}}{2 \left (g \,x^{n}+f \right ) f n}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3}}{2 \left (g \,x^{n}+f \right ) f 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 (x^{n}\right )}{2 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 (g \,x^{n}+f \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 (x^{n}\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 (g \,x^{n}+f \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 (x^{n}\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 (g \,x^{n}+f \right )}{2 f^{2} n}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (x^{n}\right )}{2 f^{2} n}+\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (g \,x^{n}+f \right )}{2 f^{2} n}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {e \,x^{n}+d}{d}\right )}{f^{2} n}+\frac {p \ln \left (\frac {d g -e f +\left (g \,x^{n}+f \right ) e}{d g -e f}\right ) \ln \left (g \,x^{n}+f \right )}{f^{2} n}+\frac {\ln \relax (c )}{\left (g \,x^{n}+f \right ) f n}+\frac {\ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{\left (g \,x^{n}+f \right ) f n}-\frac {p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{f^{2} n}+\frac {p \dilog \left (\frac {d g -e f +\left (g \,x^{n}+f \right ) e}{d g -e f}\right )}{f^{2} n}+\frac {\ln \relax (c ) \ln \left (x^{n}\right )}{f^{2} n}-\frac {\ln \relax (c ) \ln \left (g \,x^{n}+f \right )}{f^{2} n}+\frac {\ln \left (x^{n}\right ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{f^{2} n}-\frac {\ln \left (\left (e \,x^{n}+d \right )^{p}\right ) \ln \left (g \,x^{n}+f \right )}{f^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 233, normalized size = 1.14 \[ -e n p {\left (\frac {\log \left (\frac {e x^{n} + d}{e}\right )}{e f^{2} n^{2} - d f g n^{2}} - \frac {\log \left (\frac {g x^{n} + f}{g}\right )}{e f^{2} n^{2} - d f g n^{2}} + \frac {\log \left (x^{n}\right ) \log \left (\frac {e x^{n}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e x^{n}}{d}\right )}{e f^{2} n^{2}} - \frac {\log \left (g x^{n} + f\right ) \log \left (-\frac {e g x^{n} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{n} + e f}{e f - d g}\right )}{e f^{2} n^{2}}\right )} + {\left (\frac {1}{f g n x^{n} + f^{2} n} - \frac {\log \left (g x^{n} + f\right )}{f^{2} n} + \frac {\log \left (x^{n}\right )}{f^{2} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+g\,x^n\right )}^2} \,d x \]
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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