Optimal. Leaf size=127 \[ -\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}-\frac {3 \tanh ^{-1}(a x)}{10 a^5}+\frac {\coth ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\frac {3 x}{10 a^4}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^3}{30 a^2}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {x^4 \coth ^{-1}(a x)}{10 a} \]
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Rubi [A] time = 0.23, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5917, 5981, 302, 206, 321, 5985, 5919, 2402, 2315} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {3 x}{10 a^4}-\frac {3 \tanh ^{-1}(a x)}{10 a^5}+\frac {\coth ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {x^4 \coth ^{-1}(a x)}{10 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 5917
Rule 5919
Rule 5981
Rule 5985
Rubi steps
\begin {align*} \int x^4 \coth ^{-1}(a x)^2 \, dx &=\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 \tanh ^{-1}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {2 \operatorname {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 \tanh ^{-1}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 87, normalized size = 0.69 \[ \frac {6 \left (a^5 x^5-1\right ) \coth ^{-1}(a x)^2+a x \left (a^2 x^2+9\right )+3 \coth ^{-1}(a x) \left (a^4 x^4+2 a^2 x^2-4 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )-3\right )+6 \text {Li}_2\left (e^{-2 \coth ^{-1}(a x)}\right )}{30 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arcoth}\left (a x\right )^{2}, x\right ) \]
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 \[ \int x^{4} \operatorname {arcoth}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 196, normalized size = 1.54 \[ \frac {x^{5} \mathrm {arccoth}\left (a x \right )^{2}}{5}+\frac {x^{4} \mathrm {arccoth}\left (a x \right )}{10 a}+\frac {x^{2} \mathrm {arccoth}\left (a x \right )}{5 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5 a^{5}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5 a^{5}}+\frac {x^{3}}{30 a^{2}}+\frac {3 x}{10 a^{4}}+\frac {3 \ln \left (a x -1\right )}{20 a^{5}}-\frac {3 \ln \left (a x +1\right )}{20 a^{5}}+\frac {\ln \left (a x -1\right )^{2}}{20 a^{5}}-\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{5 a^{5}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{10 a^{5}}-\frac {\ln \left (a x +1\right )^{2}}{20 a^{5}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{10 a^{5}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{10 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 155, normalized size = 1.22 \[ \frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} + 18 \, a x - 3 \, \log \left (a x + 1\right )^{2} + 6 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, \log \left (a x - 1\right )^{2} + 9 \, \log \left (a x - 1\right )}{a^{7}} - \frac {12 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {9 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{10} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \operatorname {arcoth}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]
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 \[ \int x^{4} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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