Integrand size = 10, antiderivative size = 38 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{2 n} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6036, 6032} \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{2 n} \]
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Rule 6032
Rule 6036
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\operatorname {PolyLog}\left (2,-\frac {x^{-n}}{a}\right )}{2 n}-\frac {\operatorname {PolyLog}\left (2,\frac {x^{-n}}{a}\right )}{2 n} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\frac {a x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};a^2 x^{2 n}\right )}{n}+\left (\coth ^{-1}\left (a x^n\right )-\text {arctanh}\left (a x^n\right )\right ) \log (x) \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {\ln \left (a \,x^{n}-1\right ) \ln \left (a \,x^{n}\right )}{2 n}-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2 n}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2 n}\) | \(45\) |
derivativedivides | \(\frac {\ln \left (a \,x^{n}\right ) \operatorname {arccoth}\left (a \,x^{n}\right )-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\) | \(53\) |
default | \(\frac {\ln \left (a \,x^{n}\right ) \operatorname {arccoth}\left (a \,x^{n}\right )-\frac {\operatorname {dilog}\left (a \,x^{n}\right )}{2}-\frac {\operatorname {dilog}\left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.37 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac {a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - {\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) + {\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \]
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\[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {acoth}{\left (a x^{n} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.87 \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=-\frac {1}{2} \, a n {\left (\frac {\log \left (\frac {a x^{n} + 1}{a}\right )}{a n} - \frac {\log \left (\frac {a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac {1}{2} \, a n {\left (\frac {\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac {n \log \left (a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac {n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname {arcoth}\left (a x^{n}\right ) \log \left (x\right ) \]
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\[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x^{n}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acoth}\left (a\,x^n\right )}{x} \,d x \]
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