Integrand size = 21, antiderivative size = 66 \[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\frac {\text {arctanh}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {b n \operatorname {PolyLog}\left (2,-\sqrt {e} x\right )}{2 \sqrt {e}}-\frac {b n \operatorname {PolyLog}\left (2,\sqrt {e} x\right )}{2 \sqrt {e}} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {212, 2361, 12, 6031} \[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\frac {\text {arctanh}\left (\sqrt {e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {e}}+\frac {b n \operatorname {PolyLog}\left (2,-\sqrt {e} x\right )}{2 \sqrt {e}}-\frac {b n \operatorname {PolyLog}\left (2,\sqrt {e} x\right )}{2 \sqrt {e}} \]
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Rule 12
Rule 212
Rule 2361
Rule 6031
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\frac {-\left (\left (a+b \log \left (c x^n\right )\right ) \left (\log \left (1-\sqrt {e} x\right )-\log \left (1+\sqrt {e} x\right )\right )\right )+b n \operatorname {PolyLog}\left (2,-\sqrt {e} x\right )-b n \operatorname {PolyLog}\left (2,\sqrt {e} x\right )}{2 \sqrt {e}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.68 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83
method | result | size |
meijerg | \(\frac {a \,\operatorname {arctanh}\left (x \sqrt {e}\right )}{\sqrt {e}}+\frac {b \ln \left (c \right ) \operatorname {arctanh}\left (x \sqrt {e}\right )}{\sqrt {e}}+\left (\frac {b n \ln \left (x \right ) \Phi \left (e \,x^{2}, 1, \frac {1}{2}\right )}{2}-\frac {b n \Phi \left (e \,x^{2}, 2, \frac {1}{2}\right )}{4}\right ) x\) | \(55\) |
risch | \(-\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-b \ln \left (c \right )-a \right ) \operatorname {arctanh}\left (x \sqrt {e}\right )}{\sqrt {e}}-\frac {b \,\operatorname {arctanh}\left (x \sqrt {e}\right ) n \ln \left (x \right )}{\sqrt {e}}+\frac {b \,\operatorname {arctanh}\left (x \sqrt {e}\right ) \ln \left (x^{n}\right )}{\sqrt {e}}-\frac {b n \ln \left (x \right ) \ln \left (1-x \sqrt {e}\right )}{2 \sqrt {e}}+\frac {b n \ln \left (x \right ) \ln \left (x \sqrt {e}+1\right )}{2 \sqrt {e}}-\frac {b n \operatorname {dilog}\left (1-x \sqrt {e}\right )}{2 \sqrt {e}}+\frac {b n \operatorname {dilog}\left (x \sqrt {e}+1\right )}{2 \sqrt {e}}\) | \(200\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\int { -\frac {b \log \left (c x^{n}\right ) + a}{e x^{2} - 1} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=- \int \frac {a}{e x^{2} - 1}\, dx - \int \frac {b \log {\left (c x^{n} \right )}}{e x^{2} - 1}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\int { -\frac {b \log \left (c x^{n}\right ) + a}{e x^{2} - 1} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{1-e x^2} \, dx=\int -\frac {a+b\,\ln \left (c\,x^n\right )}{e\,x^2-1} \,d x \]
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