Optimal. Leaf size=204 \[ -\frac {e p \log \left (d+e x^n\right )}{f (e f-d g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {e p \log \left (f+g x^n\right )}{f (e f-d g) n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}-\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n} \]
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Rubi [A]
time = 0.18, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2525, 46,
2463, 2441, 2352, 2442, 36, 31, 2440, 2438} \begin {gather*} -\frac {p \text {PolyLog}\left (2,-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}-\frac {e p \log \left (d+e x^n\right )}{f n (e f-d g)}+\frac {e p \log \left (f+g x^n\right )}{f n (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 46
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x (f+g x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{f^2 x}-\frac {g \log \left (c (d+e x)^p\right )}{f (f+g x)^2}-\frac {g \log \left (c (d+e x)^p\right )}{f^2 (f+g x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^n\right )}{f^2 n}-\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^n\right )}{f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 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^2 n}+\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^n\right )}{f^2 n}-\frac {(e p) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^n\right )}{f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^n\right )}{f^2 n}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{f (e f-d g) n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^n\right )}{f (e f-d g) n}\\ &=-\frac {e p \log \left (d+e x^n\right )}{f (e f-d g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {e p \log \left (f+g x^n\right )}{f (e f-d g) n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}-\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 n}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 171, normalized size = 0.84 \begin {gather*} \frac {-\frac {e f p \log \left (d+e x^n\right )}{e f-d g}+\frac {f \log \left (c \left (d+e x^n\right )^p\right )}{f+g x^n}+\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {e f p \log \left (f+g x^n\right )}{e f-d g}-\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )-p \text {Li}_2\left (\frac {g \left (d+e x^n\right )}{-e f+d g}\right )+p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f^2 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 = 0.84, size = 805, normalized size = 3.95
method | result | size |
risch | \(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n f \left (f +g \,x^{n}\right )}-\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (f +g \,x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (x^{n}\right )}{n \,f^{2}}-\frac {p e \ln \left (f +g \,x^{n}\right )}{n f \left (d g -e f \right )}+\frac {p e \ln \left (d +e \,x^{n}\right )}{n f \left (d g -e f \right )}-\frac {p \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}+\frac {p \dilog \left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}+\frac {p \ln \left (f +g \,x^{n}\right ) \ln \left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}-\frac {i \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} \ln \left (f +g \,x^{n}\right )}{2 n \,f^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) \ln \left (x^{n}\right )}{2 n \,f^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 n f \left (f +g \,x^{n}\right )}-\frac {i \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 n f \left (f +g \,x^{n}\right )}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3}}{2 n f \left (f +g \,x^{n}\right )}+\frac {i \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 n f \left (f +g \,x^{n}\right )}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} \ln \left (f +g \,x^{n}\right )}{2 n \,f^{2}}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} \ln \left (x^{n}\right )}{2 n \,f^{2}}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) \ln \left (f +g \,x^{n}\right )}{2 n \,f^{2}}+\frac {i \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 ) \ln \left (f +g \,x^{n}\right )}{2 n \,f^{2}}-\frac {i \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 ) \ln \left (x^{n}\right )}{2 n \,f^{2}}+\frac {i \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} \ln \left (x^{n}\right )}{2 n \,f^{2}}+\frac {\ln \left (c \right )}{n f \left (f +g \,x^{n}\right )}-\frac {\ln \left (c \right ) \ln \left (f +g \,x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (c \right ) \ln \left (x^{n}\right )}{n \,f^{2}}\) | \(805\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 251, normalized size = 1.23 \begin {gather*} n p {\left (\frac {\log \left ({\left (d + e^{\left (n \log \left (x\right ) + 1\right )}\right )} e^{\left (-1\right )}\right )}{d f g n^{2} - f^{2} n^{2} e} - \frac {\log \left (\frac {g x^{n} + f}{g}\right )}{d f g n^{2} - f^{2} n^{2} e} + \frac {{\left (\log \left (g x^{n} + f\right ) \log \left (\frac {f e + g e^{\left (n \log \left (x\right ) + 1\right )}}{d g - f e} + 1\right ) + {\rm Li}_2\left (-\frac {f e + g e^{\left (n \log \left (x\right ) + 1\right )}}{d g - f e}\right )\right )} e^{\left (-1\right )}}{f^{2} n^{2}} - \frac {{\left (\log \left (x^{n}\right ) \log \left (\frac {e^{\left (n \log \left (x\right ) + 1\right )}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e^{\left (n \log \left (x\right ) + 1\right )}}{d}\right )\right )} e^{\left (-1\right )}}{f^{2} n^{2}}\right )} e + {\left (\frac {1}{f g n x^{n} + f^{2} n} - \frac {\log \left (g x^{n} + f\right )}{f^{2} n} + \frac {\log \left (x^{n}\right )}{f^{2} n}\right )} \log \left ({\left (x^{n} e + d\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+g\,x^n\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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