Optimal. Leaf size=41 \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)+\frac {1}{2} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5982, 5916, 266, 36, 29, 31, 5948} \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)+\frac {1}{2} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 5916
Rule 5948
Rule 5982
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{x}+\frac {1}{2} a \tanh ^{-1}(a x)^2+a \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{x}+\frac {1}{2} 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 {\tanh ^{-1}(a x)}{x}+\frac {1}{2} 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 {\tanh ^{-1}(a x)}{x}+\frac {1}{2} 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 = 0.04, size = 41, normalized size = 1.00 \[ -\frac {1}{2} a \log \left (1-a^2 x^2\right )+a \log (x)+\frac {1}{2} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 63, normalized size = 1.54 \[ \frac {a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, a x \log \left (a^{2} x^{2} - 1\right ) + 8 \, a x \log \relax (x) - 4 \, \log \left (-\frac {a x + 1}{a x - 1}\right )}{8 \, x} \]
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 \[ \int -\frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 132, normalized size = 3.22 \[ -\frac {\arctanh \left (a x \right )}{x}-\frac {a \arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {a \arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a \ln \left (a x -1\right )^{2}}{8}+\frac {a \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4}+a \ln \left (a x \right )-\frac {a \ln \left (a x -1\right )}{2}-\frac {a \ln \left (a x +1\right )}{2}-\frac {a \ln \left (a x +1\right )^{2}}{8}-\frac {a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4}+\frac {a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 82, normalized size = 2.00 \[ \frac {1}{8} \, {\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right ) + 8 \, \log \relax (x)\right )} a + \frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 80, normalized size = 1.95 \[ \frac {a\,{\ln \left (a\,x+1\right )}^2}{8}+\frac {a\,{\ln \left (1-a\,x\right )}^2}{8}-\frac {\ln \left (a\,x+1\right )}{2\,x}+\frac {\ln \left (1-a\,x\right )}{2\,x}-\frac {a\,\ln \left (a^2\,x^2-1\right )}{2}+a\,\ln \relax (x)-\frac {a\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.42, size = 37, normalized size = 0.90 \[ \begin {cases} a \log {\relax (x )} - a \log {\left (x - \frac {1}{a} \right )} + \frac {a \operatorname {atanh}^{2}{\left (a x \right )}}{2} - a \operatorname {atanh}{\left (a x \right )} - \frac {\operatorname {atanh}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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