Integrand size = 10, antiderivative size = 127 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5} \]
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Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6038, 6128, 308, 212, 327, 6132, 6056, 2449, 2352} \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {\coth ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\frac {3 x}{10 a^4}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^3}{30 a^2}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {x^4 \coth ^{-1}(a x)}{10 a} \]
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Rule 212
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6038
Rule 6056
Rule 6128
Rule 6132
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx \\ & = \frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {2 \int x^3 \coth ^{-1}(a x) \, dx}{5 a}-\frac {2 \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a} \\ & = \frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{10} \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {2 \int x \coth ^{-1}(a x) \, dx}{5 a^3}-\frac {2 \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3} \\ & = \frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{10} \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac {2 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{5 a^2} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^5} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {a x \left (9+a^2 x^2\right )+6 \left (-1+a^5 x^5\right ) \coth ^{-1}(a x)^2+3 \coth ^{-1}(a x) \left (-3+2 a^2 x^2+a^4 x^4-4 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+6 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{30 a^5} \]
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Time = 0.47 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29
method | result | size |
parts | \(\frac {x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\) | \(164\) |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\) | \(165\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\) | \(165\) |
risch | \(\frac {413}{2250 a^{5}}+\frac {x^{3}}{30 a^{2}}+\frac {47 \ln \left (a x +1\right )}{600 a^{5}}+\frac {\ln \left (a x +1\right ) x^{4}}{40 a}-\frac {\ln \left (a x +1\right ) x^{3}}{30 a^{2}}+\frac {\ln \left (a x +1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (a x +1\right ) x}{10 a^{4}}-\frac {\ln \left (a x -1\right ) x^{4}}{40 a}-\frac {\ln \left (a x -1\right ) x^{3}}{30 a^{2}}-\frac {\ln \left (a x -1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (a x -1\right ) x}{10 a^{4}}+\frac {3 x}{10 a^{4}}+\frac {137 \ln \left (a x -1\right )}{600 a^{5}}+\frac {\ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {\ln \left (a x +1\right )^{2}}{20 a^{5}}+\frac {\ln \left (a x -1\right )^{2} x^{5}}{20}-\frac {\ln \left (a x -1\right ) x^{5}}{50}-\frac {\ln \left (a x -1\right )^{2}}{20 a^{5}}-\frac {x^{5} \ln \left (a x +1\right )}{50}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {2 \left (a x -1\right )^{3} \ln \left (a x -1\right )}{15 a^{5}}-\frac {\left (\left (-\frac {1}{25}+\frac {\ln \left (a x -1\right )}{5}\right ) \left (a x -1\right )^{5}+\left (-\frac {1}{4}+\ln \left (a x -1\right )\right ) \left (a x -1\right )^{4}+\left (-\frac {2}{3}+2 \ln \left (a x -1\right )\right ) \left (a x -1\right )^{3}+\left (-1+2 \ln \left (a x -1\right )\right ) \left (a x -1\right )^{2}+\left (-1+\ln \left (a x -1\right )\right ) \left (a x -1\right )\right ) \ln \left (a x +1\right )}{2 a^{5}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )}{10 a^{5}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {3 \left (a x -1\right )^{4} \ln \left (a x -1\right )}{40 a^{5}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )^{2}}{10 a^{5}}+\frac {\left (a x -1\right )^{5} \ln \left (a x -1\right )}{50 a^{5}}\) | \(439\) |
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\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
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\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^{4} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} + 18 \, a x - 3 \, \log \left (a x + 1\right )^{2} + 6 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, \log \left (a x - 1\right )^{2} + 9 \, \log \left (a x - 1\right )}{a^{7}} - \frac {12 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {9 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{10} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \operatorname {arcoth}\left (a x\right ) \]
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\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^4\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]
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