Optimal. Leaf size=221 \[ \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {-f} x^n+\sqrt {g}\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (e x^n+d\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (e x^n+d\right )}{\sqrt {-f} d+e \sqrt {g}}\right )}{2 f n} \]
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Rubi [A] time = 0.41, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2475, 263, 260, 2416, 2394, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {-f} x^n+\sqrt {g}\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n} \]
Antiderivative was successfully verified.
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Rule 260
Rule 263
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2475
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x^2}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}-\sqrt {-f} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {g}+\sqrt {-f} x\right )}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}-\sqrt {-f} x} \, dx,x,x^n\right )}{2 \sqrt {-f} n}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {g}+\sqrt {-f} x} \, dx,x,x^n\right )}{2 \sqrt {-f} n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f n}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {g}+\sqrt {-f} x\right )}{-d \sqrt {-f}+e \sqrt {g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-f} x}{-d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f n}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-f} x}{d \sqrt {-f}+e \sqrt {g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f n}+\frac {p \text {Li}_2\left (\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f n}\\ \end {align*}
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Mathematica [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2 \, n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x x^{2 \, n} + g x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{2 \, n}}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.67, size = 461, normalized size = 2.09 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{2 n}+g \right )}{4 f n}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{2 n}+g \right )}{4 f n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \left (f \,x^{2 n}+g \right )}{4 f n}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \left (f \,x^{2 n}+g \right )}{4 f n}+\frac {p \ln \left (\frac {d f +\sqrt {-f g}\, e -\left (e \,x^{n}+d \right ) f}{d f +\sqrt {-f g}\, e}\right ) \ln \left (e \,x^{n}+d \right )}{2 f n}+\frac {p \ln \left (\frac {-d f +\sqrt {-f g}\, e +\left (e \,x^{n}+d \right ) f}{-d f +\sqrt {-f g}\, e}\right ) \ln \left (e \,x^{n}+d \right )}{2 f n}-\frac {p \ln \left (e \,x^{n}+d \right ) \ln \left (f \,x^{2 n}+g \right )}{2 f n}+\frac {p \dilog \left (\frac {d f +\sqrt {-f g}\, e -\left (e \,x^{n}+d \right ) f}{d f +\sqrt {-f g}\, e}\right )}{2 f n}+\frac {p \dilog \left (\frac {-d f +\sqrt {-f g}\, e +\left (e \,x^{n}+d \right ) f}{-d f +\sqrt {-f g}\, e}\right )}{2 f n}+\frac {\ln \relax (c ) \ln \left (f \,x^{2 n}+g \right )}{2 f n}+\frac {\ln \left (\left (e \,x^{n}+d \right )^{p}\right ) \ln \left (f \,x^{2 n}+g \right )}{2 f n} \]
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 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{2 \, n}}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,\left (f+\frac {g}{x^{2\,n}}\right )} \,d x \]
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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