Optimal. Leaf size=50 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n} \]
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Rubi [A]
time = 0.03, 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, 2504, 2441,
2352} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2352
Rule 2441
Rule 2504
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 {\text {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) \text {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 \begin {gather*} \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 {d+e x^n}{d}\right )}{f 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 = 1.45, size = 201, normalized size = 4.02
method | result | size |
risch | \(\frac {\ln \left (x \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{f}+\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2}}{2 f}-\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 f}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3}}{2 f}+\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 f}+\frac {\ln \left (c \right ) \ln \left (x \right )}{f}-\frac {p \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{f n}-\frac {p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{f}\) | \(201\) |
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 = 66, normalized size = 1.32 \begin {gather*} \frac {n p \log \left (x^{n} e + d\right ) \log \left (x\right ) - n p \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + n \log \left (c\right ) \log \left (x\right ) - p {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right )}{f 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*} \frac {\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx}{f} \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 = -0.02 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{f\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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