Optimal. Leaf size=82 \[ -\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {1}{2} a \log \left (1-a^2 x^2\right )+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a \log (x)+\frac {3}{4} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.15, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6030, 5982, 5916, 266, 36, 29, 31, 5948, 5956, 261} \[ -\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {1}{2} a \log \left (1-a^2 x^2\right )+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+a \log (x)+\frac {3}{4} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 261
Rule 266
Rule 5916
Rule 5948
Rule 5956
Rule 5982
Rule 6030
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a \tanh ^{-1}(a x)^2+a^2 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{2} a^3 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{x}+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {3}{4} a \tanh ^{-1}(a x)^2+a \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{x}+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {3}{4} a \tanh ^{-1}(a x)^2+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{x}+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {3}{4} a \tanh ^{-1}(a x)^2+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{x}+\frac {a^2 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {3}{4} a \tanh ^{-1}(a x)^2+a \log (x)-\frac {1}{2} a \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 77, normalized size = 0.94 \[ \frac {1}{4} \left (a \left (\frac {1}{a^2 x^2-1}-2 \log \left (1-a^2 x^2\right )+4 \log (a x)\right )-\frac {2 \left (3 a^2 x^2-2\right ) \tanh ^{-1}(a x)}{x \left (a^2 x^2-1\right )}+3 a \tanh ^{-1}(a x)^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 118, normalized size = 1.44 \[ \frac {3 \, {\left (a^{3} x^{3} - a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, a x - 8 \, {\left (a^{3} x^{3} - a x\right )} \log \left (a^{2} x^{2} - 1\right ) + 16 \, {\left (a^{3} x^{3} - a x\right )} \log \relax (x) - 4 \, {\left (3 \, a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{16 \, {\left (a^{2} x^{3} - x\right )}} \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 180, normalized size = 2.20 \[ -\frac {\arctanh \left (a x \right )}{x}-\frac {a \arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {3 a \arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}-\frac {a \arctanh \left (a x \right )}{4 \left (a x +1\right )}+\frac {3 a \arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {3 a \ln \left (a x -1\right )^{2}}{16}+\frac {3 a \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8}-\frac {3 a \ln \left (a x +1\right )^{2}}{16}-\frac {3 a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{8}+\frac {3 a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{8}+a \ln \left (a x \right )-\frac {a \ln \left (a x -1\right )}{2}+\frac {a}{8 a x -8}-\frac {a \ln \left (a x +1\right )}{2}-\frac {a}{8 \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 150, normalized size = 1.83 \[ -\frac {1}{16} \, a {\left (\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4}{a^{2} x^{2} - 1} + 8 \, \log \left (a x + 1\right ) + 8 \, \log \left (a x - 1\right ) - 16 \, \log \relax (x)\right )} + \frac {1}{4} \, {\left (3 \, a \log \left (a x + 1\right ) - 3 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} - 2\right )}}{a^{2} x^{3} - x}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 132, normalized size = 1.61 \[ \frac {3\,a\,{\ln \left (a\,x+1\right )}^2}{16}+\frac {3\,a\,{\ln \left (1-a\,x\right )}^2}{16}+\frac {a}{2\,\left (2\,a^2\,x^2-2\right )}-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}+a\,\ln \relax (x)-\ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,a^2\,x^2}{2}-1}{2\,x-2\,a^2\,x^3}+\frac {3\,a\,\ln \left (a\,x+1\right )}{8}\right )+\frac {\ln \left (a\,x+1\right )\,\left (\frac {3\,a\,x^2}{4}-\frac {1}{2\,a}\right )}{\frac {x}{a}-a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.72, size = 253, normalized size = 3.09 \[ \begin {cases} \frac {4 a^{3} x^{3} \log {\relax (x )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a^{3} x^{3} \log {\left (x - \frac {1}{a} \right )}}{4 a^{2} x^{3} - 4 x} + \frac {3 a^{3} x^{3} \operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a^{3} x^{3} \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {6 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} - \frac {4 a x \log {\relax (x )}}{4 a^{2} x^{3} - 4 x} + \frac {4 a x \log {\left (x - \frac {1}{a} \right )}}{4 a^{2} x^{3} - 4 x} - \frac {3 a x \operatorname {atanh}^{2}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} + \frac {4 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} + \frac {a x}{4 a^{2} x^{3} - 4 x} + \frac {4 \operatorname {atanh}{\left (a x \right )}}{4 a^{2} x^{3} - 4 x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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