Integrand size = 25, antiderivative size = 37 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \operatorname {PolyLog}\left (2,1-c x^n\right )}{c n} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2378, 2370, 2353, 2352} \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \operatorname {PolyLog}\left (2,1-c x^n\right )}{c n} \]
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Rule 2352
Rule 2353
Rule 2370
Rule 2378
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{\left (c-\frac {1}{x}\right ) x} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{-1+c x} \, dx,x,x^n\right )}{n} \\ & = \frac {a \log \left (1-c x^n\right )}{c n}+\frac {b \text {Subst}\left (\int \frac {\log (c x)}{-1+c x} \, dx,x,x^n\right )}{n} \\ & = \frac {a \log \left (1-c x^n\right )}{c n}-\frac {b \text {Li}_2\left (1-c x^n\right )}{c n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-c x^n\right )+b \operatorname {PolyLog}\left (2,c x^n\right )}{c n} \]
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Time = 0.51 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (c \,x^{n}-1\right )}{c}-\frac {b \operatorname {dilog}\left (c \,x^{n}\right )}{c}}{n}\) | \(31\) |
default | \(\frac {\frac {a \ln \left (c \,x^{n}-1\right )}{c}-\frac {b \operatorname {dilog}\left (c \,x^{n}\right )}{c}}{n}\) | \(31\) |
parts | \(\frac {a \ln \left (c \,x^{n}-1\right )}{n c}-\frac {b \operatorname {dilog}\left (c \,x^{n}\right )}{n c}\) | \(33\) |
risch | \(\frac {b \ln \left (1-c \,x^{n}\right ) \ln \left (x^{n}\right )}{n c}-\frac {b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right )}{n c}-\frac {b \operatorname {dilog}\left (c \,x^{n}\right )}{n c}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \ln \left (c \,x^{n}-1\right )}{n c}\) | \(165\) |
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Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\frac {b n \log \left (-c x^{n} + 1\right ) \log \left (x\right ) + b {\rm Li}_2\left (c x^{n}\right ) + {\left (b \log \left (c\right ) + a\right )} \log \left (c x^{n} - 1\right )}{c n} \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (c - \frac {1}{x^{n}}\right )} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (c - \frac {1}{x^{n}}\right )} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (c-x^{-n}\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (c-\frac {1}{x^n}\right )} \,d x \]
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