Optimal. Leaf size=143 \[ \frac {2}{15} a^5 \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {2}{15} a^5 \tanh ^{-1}(a x)^2-\frac {1}{30} a^5 \tanh ^{-1}(a x)-\frac {4}{15} a^5 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)+\frac {a^4}{30 x}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac {a^2}{30 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {\tanh ^{-1}(a x)^2}{5 x^5}-\frac {a \tanh ^{-1}(a x)}{10 x^4} \]
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Rubi [A] time = 0.45, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6014, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ \frac {2}{15} a^5 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {a^2}{30 x^3}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^4}{30 x}-\frac {2}{15} a^5 \tanh ^{-1}(a x)^2-\frac {1}{30} a^5 \tanh ^{-1}(a x)-\frac {4}{15} a^5 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{10 x^4}-\frac {\tanh ^{-1}(a x)^2}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 2447
Rule 5916
Rule 5932
Rule 5982
Rule 5988
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x^6} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)^2}{x^4} \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x^6} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{5} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{5} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^5} \, dx+\frac {1}{5} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{10 x^4}+\frac {a^3 \tanh ^{-1}(a x)}{3 x^2}-\frac {1}{3} a^5 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{10} a^2 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{5} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{3} a^4 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{5} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a^2}{30 x^3}+\frac {a^4}{3 x}-\frac {a \tanh ^{-1}(a x)}{10 x^4}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac {2}{15} a^5 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a^5 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{10} a^4 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{5} a^4 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{5} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{3} a^6 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{3} \left (2 a^6\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{30 x^3}+\frac {a^4}{30 x}-\frac {1}{3} a^5 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{10 x^4}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac {2}{15} a^5 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {4}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{3} a^5 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{10} a^6 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{5} a^6 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{5} \left (2 a^6\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{30 x^3}+\frac {a^4}{30 x}-\frac {1}{30} a^5 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{10 x^4}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac {2}{15} a^5 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {4}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {2}{15} a^5 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.62, size = 114, normalized size = 0.80 \[ \frac {4 a^5 x^5 \text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )+a^2 x^2 \left (a^2 x^2-1\right )-2 \left (2 a^5 x^5-5 a^2 x^2+3\right ) \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x) \left (a^4 x^4+8 a^4 x^4 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-4 a^2 x^2+3\right )}{30 x^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{6}}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 258, normalized size = 1.80 \[ \frac {a^{2} \arctanh \left (a x \right )^{2}}{3 x^{3}}-\frac {\arctanh \left (a x \right )^{2}}{5 x^{5}}-\frac {a \arctanh \left (a x \right )}{10 x^{4}}+\frac {2 a^{3} \arctanh \left (a x \right )}{15 x^{2}}-\frac {4 a^{5} \arctanh \left (a x \right ) \ln \left (a x \right )}{15}+\frac {2 a^{5} \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 a^{5} \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15}-\frac {a^{2}}{30 x^{3}}+\frac {a^{4}}{30 x}+\frac {a^{5} \ln \left (a x -1\right )}{60}-\frac {a^{5} \ln \left (a x +1\right )}{60}+\frac {2 a^{5} \dilog \left (a x \right )}{15}+\frac {2 a^{5} \dilog \left (a x +1\right )}{15}+\frac {2 a^{5} \ln \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{5} \ln \left (a x -1\right )^{2}}{30}-\frac {2 a^{5} \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{15}-\frac {a^{5} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{15}-\frac {a^{5} \ln \left (a x +1\right )^{2}}{30}+\frac {a^{5} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{15}-\frac {a^{5} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 228, normalized size = 1.59 \[ -\frac {1}{60} \, {\left (8 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a^{3} - 8 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a^{3} + 8 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a^{3} + a^{3} \log \left (a x + 1\right ) - a^{3} \log \left (a x - 1\right ) + \frac {2 \, {\left (a^{3} x^{3} \log \left (a x + 1\right )^{2} - 2 \, a^{3} x^{3} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a^{3} x^{3} \log \left (a x - 1\right )^{2} - a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a^{2} + \frac {1}{30} \, {\left (4 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{4} \log \left (x^{2}\right ) + \frac {4 \, a^{2} x^{2} - 3}{x^{4}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (5 \, a^{2} x^{2} - 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{15 \, x^{5}} \]
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 \[ -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^6} \,d x \]
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 \[ - \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{6}}\right )\, dx - \int \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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