Optimal. Leaf size=50 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{f n} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {12, 2454, 2394, 2315} \[ \frac {p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2315
Rule 2394
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx &=\frac {\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f 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 n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.92 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \text {Li}_2\left (\frac {e x^n+d}{d}\right )}{f n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 63, normalized size = 1.26 \[ \frac {n p \log \left (e x^{n} + d\right ) \log \relax (x) - n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) + n \log \relax (c) \log \relax (x) - p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{f n} \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.79, size = 201, normalized size = 4.02 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \relax (x )}{2 f}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2 f}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2 f}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \relax (x )}{2 f}-\frac {p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )}{f}+\frac {\ln \relax (c ) \ln \relax (x )}{f}+\frac {\ln \relax (x ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{f}-\frac {p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{f n} \]
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 \[ \frac {2 \, d n p \int \frac {\log \relax (x)}{e x x^{n} + d x}\,{d x} - n p \log \relax (x)^{2} + 2 \, \log \left ({\left (e x^{n} + d\right )}^{p}\right ) \log \relax (x) + 2 \, \log \relax (c) \log \relax (x)}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{f\,x} \,d x \]
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 \[ \frac {\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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