Optimal. Leaf size=66 \[ \frac {\tanh ^{-1}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {b n \text {Li}_2\left (-\sqrt {e} x\right )}{2 \sqrt {e}}-\frac {b n \text {Li}_2\left (\sqrt {e} x\right )}{2 \sqrt {e}} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {212, 2361, 12,
6031} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\sqrt {e} x\right )}{2 \sqrt {e}}-\frac {b n \text {PolyLog}\left (2,\sqrt {e} x\right )}{2 \sqrt {e}}+\frac {\tanh ^{-1}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 2361
Rule 6031
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx &=\frac {\tanh ^{-1}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-(b n) \int \frac {\tanh ^{-1}\left (\sqrt {e} x\right )}{\sqrt {e} x} \, dx\\ &=\frac {\tanh ^{-1}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}-\frac {(b n) \int \frac {\tanh ^{-1}\left (\sqrt {e} x\right )}{x} \, dx}{\sqrt {e}}\\ &=\frac {\tanh ^{-1}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {b n \text {Li}_2\left (-\sqrt {e} x\right )}{2 \sqrt {e}}-\frac {b n \text {Li}_2\left (\sqrt {e} x\right )}{2 \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 72, normalized size = 1.09 \begin {gather*} \frac {-\left (\left (a+b \log \left (c x^n\right )\right ) \left (\log \left (1-\sqrt {e} x\right )-\log \left (1+\sqrt {e} x\right )\right )\right )+b n \text {Li}_2\left (-\sqrt {e} x\right )-b n \text {Li}_2\left (\sqrt {e} x\right )}{2 \sqrt {e}} \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 = 0.13, size = 55, normalized size = 0.83
method | result | size |
meijerg | \(\frac {a \arctanh \left (x \sqrt {e}\right )}{\sqrt {e}}+\frac {b \ln \left (c \right ) \arctanh \left (x \sqrt {e}\right )}{\sqrt {e}}+\left (\frac {b n \ln \left (x \right ) \Phi \left (e \,x^{2}, 1, \frac {1}{2}\right )}{2}-\frac {b n \Phi \left (e \,x^{2}, 2, \frac {1}{2}\right )}{4}\right ) x\) | \(55\) |
risch | \(-\frac {\left (\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}-b \ln \left (c \right )-a \right ) \arctanh \left (x \sqrt {e}\right )}{\sqrt {e}}-\frac {b \arctanh \left (x \sqrt {e}\right ) n \ln \left (x \right )}{\sqrt {e}}+\frac {b \arctanh \left (x \sqrt {e}\right ) \ln \left (x^{n}\right )}{\sqrt {e}}-\frac {b n \ln \left (x \right ) \ln \left (1-x \sqrt {e}\right )}{2 \sqrt {e}}+\frac {b n \ln \left (x \right ) \ln \left (x \sqrt {e}+1\right )}{2 \sqrt {e}}-\frac {b n \dilog \left (1-x \sqrt {e}\right )}{2 \sqrt {e}}+\frac {b n \dilog \left (x \sqrt {e}+1\right )}{2 \sqrt {e}}\) | \(200\) |
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 \begin {gather*} \text {Failed to integrate} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {a}{e x^{2} - 1}\, dx - \int \frac {b \log {\left (c x^{n} \right )}}{e x^{2} - 1}\, dx \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 = -0.02 \begin {gather*} \int -\frac {a+b\,\ln \left (c\,x^n\right )}{e\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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