Optimal. Leaf size=124 \[ \frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{1+n} (f x)^{-1-n} \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{d n} \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2506, 2504,
2444, 2441, 2352} \begin {gather*} \frac {2 e p^2 x^{n+1} (f x)^{-n-1} \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{d n}-\frac {x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2444
Rule 2504
Rule 2506
Rubi steps
\begin {align*} \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx &=\left (x^{1+n} (f x)^{-1-n}\right ) \int x^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\\ &=\frac {\left (x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {\left (2 e p x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{d n}\\ &=\frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {\left (2 e^2 p^2 x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{d n}\\ &=\frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{1+n} (f x)^{-1-n} \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{d n}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 148, normalized size = 1.19 \begin {gather*} -\frac {(f x)^{-n} \left (2 e p^2 x^n \log \left (-\frac {d x^{-n}}{e}\right ) \log \left (-e-d x^{-n}\right )-e p^2 x^n \log ^2\left (-e-d x^{-n}\right )+2 e p x^n \log \left (-e-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+d \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e p^2 x^n \text {Li}_2\left (1+\frac {d x^{-n}}{e}\right )\right )}{d f n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{-1-n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 207, normalized size = 1.67 \begin {gather*} -\frac {2 \, f^{-n - 1} n p^{2} x^{n} e \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) - 2 \, f^{-n - 1} n p x^{n} e \log \left (c\right ) \log \left (x\right ) + 2 \, f^{-n - 1} p^{2} x^{n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) e + d f^{-n - 1} \log \left (c\right )^{2} + {\left (f^{-n - 1} p^{2} x^{n} e + d f^{-n - 1} p^{2}\right )} \log \left (x^{n} e + d\right )^{2} + 2 \, {\left (d f^{-n - 1} p \log \left (c\right ) - {\left (n p^{2} e \log \left (x\right ) - p e \log \left (c\right )\right )} f^{-n - 1} x^{n}\right )} \log \left (x^{n} e + d\right )}{d n x^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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