Optimal. Leaf size=124 \[ -\frac {x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{n+1} (f x)^{-n-1} \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{d n} \]
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Rubi [A] time = 0.11, 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 = {2456, 2454, 2397, 2394, 2315} \[ \frac {2 e p^2 x^{n+1} (f x)^{-n-1} \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{d n}-\frac {x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2397
Rule 2454
Rule 2456
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.09, size = 148, normalized size = 1.19 \[ -\frac {(f x)^{-n} \left (d \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e p x^n \log \left (-d x^{-n}-e\right ) \log \left (c \left (d+e x^n\right )^p\right )+2 e p^2 x^n \text {Li}_2\left (\frac {d x^{-n}}{e}+1\right )-e p^2 x^n \log ^2\left (-d x^{-n}-e\right )+2 e p^2 x^n \log \left (-\frac {d x^{-n}}{e}\right ) \log \left (-d x^{-n}-e\right )\right )}{d f n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 197, normalized size = 1.59 \[ -\frac {2 \, e f^{-n - 1} n p^{2} x^{n} \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - 2 \, e f^{-n - 1} n p x^{n} \log \relax (c) \log \relax (x) + 2 \, e f^{-n - 1} p^{2} x^{n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d f^{-n - 1} \log \relax (c)^{2} + {\left (e f^{-n - 1} p^{2} x^{n} + d f^{-n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 2 \, {\left (d f^{-n - 1} p \log \relax (c) - {\left (e n p^{2} \log \relax (x) - e p \log \relax (c)\right )} f^{-n - 1} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
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 \[ \int \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.81, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{-n -1} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )^{2}\, dx \]
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 \[ -\frac {{\left (e n^{2} p^{2} x^{n} \log \relax (x)^{2} - e p^{2} x^{n} \log \left (e x^{n} + d\right )^{2} + d \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{2} + d \log \relax (c)^{2} - 2 \, {\left (e n p x^{n} \log \relax (x) - e p x^{n} \log \left (e x^{n} + d\right ) - d \log \relax (c)\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )\right )} f^{-n - 1}}{d n x^{n}} + \int \frac {2 \, {\left (e n p^{2} \log \relax (x) + e p \log \relax (c)\right )}}{e f^{n + 1} x x^{n} + d f^{n + 1} x}\,{d x} \]
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 \[ \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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