Optimal. Leaf size=89 \[ -\frac {1}{3} a^4 \log (x)+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {a^2}{12 x^2}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^4 \log \left (1-a^2 x^2\right )-\frac {a \tanh ^{-1}(a x)}{6 x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6008, 6014, 5916, 266, 44, 36, 29, 31} \[ -\frac {a^2}{12 x^2}+\frac {1}{6} a^4 \log \left (1-a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x)+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {a \tanh ^{-1}(a x)}{6 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 5916
Rule 6008
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x^5} \, dx &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x^4} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx-\frac {1}{2} a^3 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{6} a^2 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac {1}{2} a^4 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}+\frac {1}{12} a^2 \operatorname {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^4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{4} a^6 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {a^3 \tanh ^{-1}(a x)}{2 x}-\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x)+\frac {1}{6} a^4 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.92 \[ \frac {-4 a^4 x^4 \log (x)+\left (6 a^3 x^3-2 a x\right ) \tanh ^{-1}(a x)-a^2 x^2-3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2+2 a^4 x^4 \log \left (1-a^2 x^2\right )}{12 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 108, normalized size = 1.21 \[ \frac {8 \, a^{4} x^{4} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{4} x^{4} \log \relax (x) - 4 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{3} x^{3} - a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 282, normalized size = 3.17 \[ -\frac {1}{3} \, {\left (a^{3} \log \left (-\frac {a x + 1}{a x - 1} - 1\right ) - a^{3} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {3 \, {\left (a x + 1\right )}^{2} a^{3} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x - 1\right )}^{2} {\left (\frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {4 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} - \frac {{\left (a x + 1\right )} a^{3}}{{\left (a x - 1\right )} {\left (\frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {2 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 199, normalized size = 2.24 \[ \frac {a^{2} \arctanh \left (a x \right )^{2}}{2 x^{2}}-\frac {\arctanh \left (a x \right )^{2}}{4 x^{4}}+\frac {a^{3} \arctanh \left (a x \right )}{2 x}-\frac {a \arctanh \left (a x \right )}{6 x^{3}}+\frac {a^{4} \arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {a^{4} \arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}+\frac {a^{4} \ln \left (a x -1\right )^{2}}{16}-\frac {a^{4} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8}+\frac {a^{4} \ln \left (a x +1\right )^{2}}{16}+\frac {a^{4} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8}-\frac {a^{4} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{8}-\frac {a^{2}}{12 x^{2}}-\frac {a^{4} \ln \left (a x \right )}{3}+\frac {a^{4} \ln \left (a x -1\right )}{6}+\frac {a^{4} \ln \left (a x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 164, normalized size = 1.84 \[ -\frac {1}{48} \, {\left (16 \, a^{2} \log \relax (x) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 8 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 4 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) - 4}{x^{2}}\right )} a^{2} - \frac {1}{12} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 246, normalized size = 2.76 \[ {\ln \left (1-a\,x\right )}^2\,\left (\frac {\frac {a^2\,x^2}{2}-\frac {1}{4}}{4\,x^4}-\frac {a^4}{16}\right )-\ln \left (1-a\,x\right )\,\left (\ln \left (a\,x+1\right )\,\left (\frac {\frac {a^2\,x^2}{2}-\frac {1}{4}}{2\,x^4}-\frac {a^4}{8}\right )+\frac {3\,a^5\,x-2\,a^4}{24\,a^3\,x^3}-\frac {3\,x\,a^5+2\,a^4}{24\,a^3\,x^3}-\frac {a\,\left (22\,a^3\,x^3-12\,a^2\,x^2+6\,a\,x-4\right )}{96\,x^3}+\frac {a\,\left (44\,a^3\,x^3+24\,a^2\,x^2+12\,a\,x+8\right )}{192\,x^3}\right )-\frac {a^4\,\ln \relax (x)}{3}+{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {a^2\,x^2}{8}-\frac {1}{16}}{x^4}-\frac {a^4}{16}\right )+\frac {a^4\,\ln \left (a^2\,x^2-1\right )}{6}-\frac {a^2}{12\,x^2}+\frac {a\,\ln \left (a\,x+1\right )\,\left (\frac {a^2\,x^2}{4}-\frac {1}{12}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 102, normalized size = 1.15 \[ \begin {cases} - \frac {a^{4} \log {\relax (x )}}{3} + \frac {a^{4} \log {\left (x - \frac {1}{a} \right )}}{3} - \frac {a^{4} \operatorname {atanh}^{2}{\left (a x \right )}}{4} + \frac {a^{4} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {a^{3} \operatorname {atanh}{\left (a x \right )}}{2 x} + \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{2 x^{2}} - \frac {a^{2}}{12 x^{2}} - \frac {a \operatorname {atanh}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{4 x^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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