Optimal. Leaf size=79 \[ \frac {2 p \text {Li}_2\left (\frac {e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}-\frac {2 p^2 \text {Li}_3\left (\frac {e x^n}{d}+1\right )}{n} \]
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Rubi [A] time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2454, 2396, 2433, 2374, 6589} \[ \frac {2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {2 p^2 \text {PolyLog}\left (3,\frac {e x^n}{d}+1\right )}{n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2433
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(2 e p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(2 p) \operatorname {Subst}\left (\int \frac {\log \left (c x^p\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 p \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {\left (2 p^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 p \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {2 p^2 \text {Li}_3\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 164, normalized size = 2.08 \[ 2 p \left (\log (x) \left (\log \left (d+e x^n\right )-\log \left (\frac {e x^n}{d}+1\right )\right )-\frac {\text {Li}_2\left (-\frac {e x^n}{d}\right )}{n}\right ) \left (\log \left (c \left (d+e x^n\right )^p\right )-p \log \left (d+e x^n\right )\right )+\log (x) \left (\log \left (c \left (d+e x^n\right )^p\right )-p \log \left (d+e x^n\right )\right )^2+\frac {p^2 \left (-2 \text {Li}_3\left (\frac {e x^n}{d}+1\right )+2 \text {Li}_2\left (\frac {e x^n}{d}+1\right ) \log \left (d+e x^n\right )+\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.15, size = 1356, normalized size = 17.16 \[ \text {result too large to display} \]
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 \[ \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{2} \log \relax (x) - \int -\frac {e x^{n} \log \relax (c)^{2} + d \log \relax (c)^{2} - 2 \, {\left ({\left (e n p \log \relax (x) - e \log \relax (c)\right )} x^{n} - d \log \relax (c)\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )}{e x x^{n} + d x}\,{d x} \]
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 \[ \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{x} \,d x \]
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 \[ \int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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