Optimal. Leaf size=183 \[ -\frac {8}{105} a^7 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {1}{210} a^7 \tanh ^{-1}(a x)+\frac {16}{105} a^7 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)+\frac {a^6}{210 x}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {17 a^4}{630 x^3}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {a^2}{105 x^5}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {\tanh ^{-1}(a x)^2}{7 x^7}-\frac {a \tanh ^{-1}(a x)}{21 x^6} \]
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Rubi [A] time = 0.82, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6012, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ -\frac {8}{105} a^7 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {17 a^4}{630 x^3}-\frac {a^2}{105 x^5}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}+\frac {a^6}{210 x}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {1}{210} a^7 \tanh ^{-1}(a x)+\frac {16}{105} a^7 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}-\frac {\tanh ^{-1}(a x)^2}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 2447
Rule 5916
Rule 5932
Rule 5982
Rule 5988
Rule 6012
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^8} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x^8}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^6}+\frac {a^4 \tanh ^{-1}(a x)^2}{x^4}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^6} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)^2}{x^4} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^8} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{7} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^7 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{7} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^7} \, dx+\frac {1}{7} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5} \, dx+\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{5} \left (4 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {a^3 \tanh ^{-1}(a x)}{5 x^4}-\frac {a^5 \tanh ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{21} a^2 \int \frac {1}{x^6 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^5} \, dx-\frac {1}{5} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{5} \left (4 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {a^4}{15 x^3}-\frac {a^6}{3 x}-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}+\frac {a^5 \tanh ^{-1}(a x)}{15 x^2}-\frac {1}{15} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{21} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{14} a^4 \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx-\frac {1}{5} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (2 a^6\right ) \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{5} \left (4 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{3} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {4 a^6}{15 x}+\frac {1}{3} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}-\frac {2}{15} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{21} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{14} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} a^6 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{7} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{5} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{5} \left (2 a^8\right ) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{5} \left (4 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {a^6}{210 x}-\frac {4}{15} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {16}{105} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{15} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{21} a^8 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{14} a^8 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{7} a^8 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{7} \left (2 a^8\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{105 x^5}+\frac {17 a^4}{630 x^3}+\frac {a^6}{210 x}-\frac {1}{210} a^7 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{21 x^6}+\frac {9 a^3 \tanh ^{-1}(a x)}{70 x^4}-\frac {8 a^5 \tanh ^{-1}(a x)}{105 x^2}+\frac {8}{105} a^7 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{7 x^7}+\frac {2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}-\frac {a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac {16}{105} a^7 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {8}{105} a^7 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 1.46, size = 140, normalized size = 0.77 \[ \frac {-48 a^7 x^7 \text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )+a^2 x^2 \left (3 a^4 x^4+17 a^2 x^2-6\right )+6 \left (8 a^7 x^7-35 a^4 x^4+42 a^2 x^2-15\right ) \tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x) \left (-a^6 x^6+32 a^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-16 a^4 x^4+27 a^2 x^2-10\right )}{630 x^7} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{8}}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 292, normalized size = 1.60 \[ -\frac {a^{4} \arctanh \left (a x \right )^{2}}{3 x^{3}}-\frac {\arctanh \left (a x \right )^{2}}{7 x^{7}}+\frac {2 a^{2} \arctanh \left (a x \right )^{2}}{5 x^{5}}-\frac {a \arctanh \left (a x \right )}{21 x^{6}}+\frac {9 a^{3} \arctanh \left (a x \right )}{70 x^{4}}-\frac {8 a^{5} \arctanh \left (a x \right )}{105 x^{2}}+\frac {16 a^{7} \arctanh \left (a x \right ) \ln \left (a x \right )}{105}-\frac {8 a^{7} \arctanh \left (a x \right ) \ln \left (a x -1\right )}{105}-\frac {8 a^{7} \arctanh \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {8 a^{7} \dilog \left (a x \right )}{105}-\frac {8 a^{7} \dilog \left (a x +1\right )}{105}-\frac {8 a^{7} \ln \left (a x \right ) \ln \left (a x +1\right )}{105}-\frac {2 a^{7} \ln \left (a x -1\right )^{2}}{105}+\frac {8 a^{7} \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{105}+\frac {4 a^{7} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{105}+\frac {2 a^{7} \ln \left (a x +1\right )^{2}}{105}+\frac {4 a^{7} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{105}-\frac {4 a^{7} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{105}+\frac {a^{6}}{210 x}-\frac {a^{2}}{105 x^{5}}+\frac {17 a^{4}}{630 x^{3}}+\frac {a^{7} \ln \left (a x -1\right )}{420}-\frac {a^{7} \ln \left (a x +1\right )}{420} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 254, normalized size = 1.39 \[ \frac {1}{1260} \, {\left (96 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a^{5} - 96 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a^{5} + 96 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a^{5} - 3 \, a^{5} \log \left (a x + 1\right ) + 3 \, a^{5} \log \left (a x - 1\right ) + \frac {2 \, {\left (12 \, a^{5} x^{5} \log \left (a x + 1\right )^{2} - 24 \, a^{5} x^{5} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 12 \, a^{5} x^{5} \log \left (a x - 1\right )^{2} + 3 \, a^{4} x^{4} + 17 \, a^{2} x^{2} - 6\right )}}{x^{5}}\right )} a^{2} - \frac {1}{210} \, {\left (16 \, a^{6} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{6} \log \left (x^{2}\right ) + \frac {16 \, a^{4} x^{4} - 27 \, a^{2} x^{2} + 10}{x^{6}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (35 \, a^{4} x^{4} - 42 \, a^{2} x^{2} + 15\right )} \operatorname {artanh}\left (a x\right )^{2}}{105 \, x^{7}} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^8} \,d x \]
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 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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