Optimal. Leaf size=170 \[ -\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {4}{63} a^8 \log (x)-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right ) \]
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Rubi [A]
time = 0.61, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 56, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6159, 6037,
6129, 272, 46, 36, 29, 31, 6095} \begin {gather*} \frac {4}{63} a^8 \log (x)+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {5 a^6}{504 x^2}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac {a^4}{84 x^4}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^2}{168 x^6}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)^2}{8 x^8}-\frac {a \tanh ^{-1}(a x)}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 6037
Rule 6095
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^9} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x^9}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^7}+\frac {a^4 \tanh ^{-1}(a x)^2}{x^5}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^7} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)^2}{x^5} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^9} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{4} a \int \frac {\tanh ^{-1}(a x)}{x^8 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{4} a \int \frac {\tanh ^{-1}(a x)}{x^8} \, dx+\frac {1}{4} a^3 \int \frac {\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^6} \, dx+\frac {1}{2} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {2 a^3 \tanh ^{-1}(a x)}{15 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{6 x^3}-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{28} a^2 \int \frac {1}{x^7 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^3 \int \frac {\tanh ^{-1}(a x)}{x^6} \, dx-\frac {1}{15} \left (2 a^4\right ) \int \frac {1}{x^5 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx+\frac {1}{6} a^6 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx-\frac {1}{3} \left (2 a^7\right ) \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^9 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac {a^5 \tanh ^{-1}(a x)}{18 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{2 x}+\frac {1}{4} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{56} a^2 \text {Subst}\left (\int \frac {1}{x^4 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} a^4 \int \frac {1}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac {1}{15} a^4 \text {Subst}\left (\int \frac {1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^5 \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx+\frac {1}{12} a^6 \text {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{9} \left (2 a^6\right ) \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^7 \int \frac {\tanh ^{-1}(a x)}{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^2} \, dx+\frac {1}{2} a^8 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{3} \left (2 a^9\right ) \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac {a^7 \tanh ^{-1}(a x)}{6 x}-\frac {1}{12} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{56} a^2 \text {Subst}\left (\int \left (\frac {1}{x^4}+\frac {a^2}{x^3}+\frac {a^4}{x^2}+\frac {a^6}{x}-\frac {a^8}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{40} a^4 \text {Subst}\left (\int \frac {1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{15} a^4 \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a^2}{x^2}+\frac {a^4}{x}-\frac {a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} a^6 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{12} a^6 \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{9} a^6 \text {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^7 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx+\frac {1}{4} a^8 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^8\right ) \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^9 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{168 x^6}+\frac {41 a^4}{1680 x^4}-\frac {29 a^6}{840 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {29}{420} a^8 \log (x)-\frac {29}{840} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{40} a^4 \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {a^2}{x^2}+\frac {a^4}{x}-\frac {a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{24} a^6 \text {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{9} a^6 \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} a^8 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} a^8 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} a^8 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{4} a^{10} \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {13 a^6}{252 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {25}{63} a^8 \log (x)-\frac {25}{126} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{24} a^6 \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{8} a^8 \text {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{3} a^8 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} a^{10} \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}-\frac {47}{252} a^8 \log (x)+\frac {47}{504} a^8 \log \left (1-a^2 x^2\right )+\frac {1}{8} a^8 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{8} a^{10} \text {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{168 x^6}+\frac {a^4}{84 x^4}+\frac {5 a^6}{504 x^2}-\frac {a \tanh ^{-1}(a x)}{28 x^7}+\frac {a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac {a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac {a^7 \tanh ^{-1}(a x)}{12 x}+\frac {1}{24} a^8 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{8 x^8}+\frac {a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac {a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {4}{63} a^8 \log (x)-\frac {2}{63} a^8 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 124, normalized size = 0.73 \begin {gather*} \frac {-2 a x \left (9-21 a^2 x^2+7 a^4 x^4+21 a^6 x^6\right ) \tanh ^{-1}(a x)+21 \left (-1+a^2 x^2\right )^3 \left (3+a^2 x^2\right ) \tanh ^{-1}(a x)^2+a^2 x^2 \left (-3+6 a^2 x^2+5 a^4 x^4+32 a^6 x^6 \log (x)-16 a^6 x^6 \log \left (1-a^2 x^2\right )\right )}{504 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 226, normalized size = 1.33
method | result | size |
derivativedivides | \(a^{8} \left (\frac {\arctanh \left (a x \right )^{2}}{3 a^{6} x^{6}}-\frac {\arctanh \left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\arctanh \left (a x \right )^{2}}{8 a^{8} x^{8}}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{24}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{24}-\frac {\arctanh \left (a x \right )}{28 a^{7} x^{7}}+\frac {\arctanh \left (a x \right )}{12 a^{5} x^{5}}-\frac {\arctanh \left (a x \right )}{36 a^{3} x^{3}}-\frac {\arctanh \left (a x \right )}{12 a x}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (a x -1\right )^{2}}{96}-\frac {\ln \left (a x +1\right )^{2}}{96}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {2 \ln \left (a x +1\right )}{63}-\frac {2 \ln \left (a x -1\right )}{63}-\frac {1}{168 a^{6} x^{6}}+\frac {1}{84 a^{4} x^{4}}+\frac {5}{504 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{63}\right )\) | \(226\) |
default | \(a^{8} \left (\frac {\arctanh \left (a x \right )^{2}}{3 a^{6} x^{6}}-\frac {\arctanh \left (a x \right )^{2}}{4 a^{4} x^{4}}-\frac {\arctanh \left (a x \right )^{2}}{8 a^{8} x^{8}}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{24}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{24}-\frac {\arctanh \left (a x \right )}{28 a^{7} x^{7}}+\frac {\arctanh \left (a x \right )}{12 a^{5} x^{5}}-\frac {\arctanh \left (a x \right )}{36 a^{3} x^{3}}-\frac {\arctanh \left (a x \right )}{12 a x}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {\ln \left (a x -1\right )^{2}}{96}-\frac {\ln \left (a x +1\right )^{2}}{96}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{48}-\frac {2 \ln \left (a x +1\right )}{63}-\frac {2 \ln \left (a x -1\right )}{63}-\frac {1}{168 a^{6} x^{6}}+\frac {1}{84 a^{4} x^{4}}+\frac {5}{504 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{63}\right )\) | \(226\) |
risch | \(\frac {\left (a^{8} x^{8}-6 a^{4} x^{4}+8 a^{2} x^{2}-3\right ) \ln \left (a x +1\right )^{2}}{96 x^{8}}-\frac {\left (21 a^{8} x^{8} \ln \left (-a x +1\right )+42 a^{7} x^{7}+14 a^{5} x^{5}-126 x^{4} \ln \left (-a x +1\right ) a^{4}-42 a^{3} x^{3}+168 x^{2} \ln \left (-a x +1\right ) a^{2}+18 a x -63 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{1008 x^{8}}+\frac {21 a^{8} x^{8} \ln \left (-a x +1\right )^{2}+128 \ln \left (x \right ) a^{8} x^{8}-64 \ln \left (a^{2} x^{2}-1\right ) a^{8} x^{8}+84 a^{7} x^{7} \ln \left (-a x +1\right )+20 a^{6} x^{6}+28 x^{5} \ln \left (-a x +1\right ) a^{5}-126 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+24 a^{4} x^{4}-84 a^{3} x^{3} \ln \left (-a x +1\right )+168 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-12 a^{2} x^{2}+36 a x \ln \left (-a x +1\right )-63 \ln \left (-a x +1\right )^{2}}{2016 x^{8}}\) | \(310\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 204, normalized size = 1.20 \begin {gather*} \frac {1}{2016} \, {\left (128 \, a^{6} \log \left (x\right ) - \frac {21 \, a^{6} x^{6} \log \left (a x + 1\right )^{2} + 21 \, a^{6} x^{6} \log \left (a x - 1\right )^{2} + 64 \, a^{6} x^{6} \log \left (a x - 1\right ) - 20 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 2 \, {\left (21 \, a^{6} x^{6} \log \left (a x - 1\right ) - 32 \, a^{6} x^{6}\right )} \log \left (a x + 1\right ) + 12}{x^{6}}\right )} a^{2} + \frac {1}{504} \, {\left (21 \, a^{7} \log \left (a x + 1\right ) - 21 \, a^{7} \log \left (a x - 1\right ) - \frac {2 \, {\left (21 \, a^{6} x^{6} + 7 \, a^{4} x^{4} - 21 \, a^{2} x^{2} + 9\right )}}{x^{7}}\right )} a \operatorname {artanh}\left (a x\right ) - \frac {{\left (6 \, a^{4} x^{4} - 8 \, a^{2} x^{2} + 3\right )} \operatorname {artanh}\left (a x\right )^{2}}{24 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 148, normalized size = 0.87 \begin {gather*} -\frac {64 \, a^{8} x^{8} \log \left (a^{2} x^{2} - 1\right ) - 128 \, a^{8} x^{8} \log \left (x\right ) - 20 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 12 \, a^{2} x^{2} - 21 \, {\left (a^{8} x^{8} - 6 \, a^{4} x^{4} + 8 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (21 \, a^{7} x^{7} + 7 \, a^{5} x^{5} - 21 \, a^{3} x^{3} + 9 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2016 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.93, size = 168, normalized size = 0.99 \begin {gather*} \begin {cases} \frac {4 a^{8} \log {\left (x \right )}}{63} - \frac {4 a^{8} \log {\left (x - \frac {1}{a} \right )}}{63} + \frac {a^{8} \operatorname {atanh}^{2}{\left (a x \right )}}{24} - \frac {4 a^{8} \operatorname {atanh}{\left (a x \right )}}{63} - \frac {a^{7} \operatorname {atanh}{\left (a x \right )}}{12 x} + \frac {5 a^{6}}{504 x^{2}} - \frac {a^{5} \operatorname {atanh}{\left (a x \right )}}{36 x^{3}} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4 x^{4}} + \frac {a^{4}}{84 x^{4}} + \frac {a^{3} \operatorname {atanh}{\left (a x \right )}}{12 x^{5}} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{3 x^{6}} - \frac {a^{2}}{168 x^{6}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{28 x^{7}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{8 x^{8}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 651 vs.
\(2 (144) = 288\).
time = 0.41, size = 651, normalized size = 3.83 \begin {gather*} \frac {2}{63} \, {\left (2 \, a^{7} \log \left (-\frac {a x + 1}{a x - 1} - 1\right ) - 2 \, a^{7} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {84 \, {\left (\frac {{\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} - \frac {{\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {{\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{8}}{{\left (a x - 1\right )}^{8}} + \frac {8 \, {\left (a x + 1\right )}^{7}}{{\left (a x - 1\right )}^{7}} + \frac {28 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {56 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {70 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {56 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {28 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1} + \frac {2 \, {\left (\frac {28 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {7 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{7}}{{\left (a x - 1\right )}^{7}} + \frac {7 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {21 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {35 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {35 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {7 \, {\left (a x + 1\right )}}{a x - 1} + 1} - \frac {\frac {2 \, {\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} + \frac {11 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {6 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {11 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )} a^{7}}{a x - 1}}{\frac {{\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {6 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {20 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {6 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.72, size = 357, normalized size = 2.10 \begin {gather*} \frac {4\,a^8\,\ln \left (x\right )}{63}+\frac {a^8\,{\ln \left (a\,x+1\right )}^2}{96}+\frac {a^8\,{\ln \left (1-a\,x\right )}^2}{96}-\frac {{\ln \left (a\,x+1\right )}^2}{32\,x^8}-\frac {{\ln \left (1-a\,x\right )}^2}{32\,x^8}-\frac {2\,a^8\,\ln \left (a^2\,x^2-1\right )}{63}-\frac {a^2}{168\,x^6}+\frac {a^4}{84\,x^4}+\frac {5\,a^6}{504\,x^2}-\frac {a^8\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{48}+\frac {\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{16\,x^8}+\frac {a^2\,{\ln \left (a\,x+1\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (a\,x+1\right )}^2}{16\,x^4}+\frac {a^2\,{\ln \left (1-a\,x\right )}^2}{12\,x^6}-\frac {a^4\,{\ln \left (1-a\,x\right )}^2}{16\,x^4}-\frac {a\,\ln \left (a\,x+1\right )}{56\,x^7}+\frac {a\,\ln \left (1-a\,x\right )}{56\,x^7}+\frac {a^3\,\ln \left (a\,x+1\right )}{24\,x^5}-\frac {a^5\,\ln \left (a\,x+1\right )}{72\,x^3}-\frac {a^7\,\ln \left (a\,x+1\right )}{24\,x}-\frac {a^3\,\ln \left (1-a\,x\right )}{24\,x^5}+\frac {a^5\,\ln \left (1-a\,x\right )}{72\,x^3}+\frac {a^7\,\ln \left (1-a\,x\right )}{24\,x}-\frac {a^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{6\,x^6}+\frac {a^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{8\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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