Optimal. Leaf size=70 \[ \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f n}+\frac {p \text {Li}_2\left (\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f n} \]
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Rubi [A] time = 0.16, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2475, 2412, 2394, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f n} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2412
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 )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x}\right ) x} \, 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 )}{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 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 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 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 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 n}+\frac {p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 64, normalized size = 0.91 \[ \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f x^n+g\right )}{e g-d f}\right )+p \text {Li}_2\left (\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x x^{n} + g x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 298, normalized size = 4.26 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{n}+g \right )}{2 f n}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{n}+g \right )}{2 f n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{n}+g \right )}{2 f n}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (f \,x^{n}+g \right )}{2 f n}-\frac {p \ln \left (\frac {d f -e g +\left (f \,x^{n}+g \right ) e}{d f -e g}\right ) \ln \left (f \,x^{n}+g \right )}{f n}-\frac {p \dilog \left (\frac {d f -e g +\left (f \,x^{n}+g \right ) e}{d f -e g}\right )}{f n}+\frac {\ln \relax (c ) \ln \left (f \,x^{n}+g \right )}{f n}+\frac {\ln \left (\left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{n}+g \right )}{f n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 112, normalized size = 1.60 \[ {\left (\frac {\log \left (f + \frac {g}{x^{n}}\right )}{f n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) - \frac {{\left (\log \left (f x^{n} + g\right ) \log \left (\frac {e f x^{n} + e g}{d f - e g} + 1\right ) + {\rm Li}_2\left (-\frac {e f x^{n} + e g}{d f - e g}\right )\right )} p}{f n} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,\left (f+\frac {g}{x^n}\right )} \,d x \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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