Optimal. Leaf size=88 \[ \frac {x}{4 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}+\frac {\tanh ^{-1}(a x)}{4 a} \]
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Rubi [A] time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5956, 5994, 199, 206} \[ \frac {x}{4 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}+\frac {\tanh ^{-1}(a x)}{4 a} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5956
Rule 5994
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}-a \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}+\frac {1}{2} \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {x}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}+\frac {1}{4} \int \frac {1}{1-a^2 x^2} \, dx\\ &=\frac {x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{4 a}-\frac {\tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{6 a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 93, normalized size = 1.06 \[ \frac {-3 \left (\left (a^2 x^2-1\right ) \log (1-a x)+\left (1-a^2 x^2\right ) \log (a x+1)+2 a x\right )+4 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^3-12 a x \tanh ^{-1}(a x)^2+12 \tanh ^{-1}(a x)}{24 a \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 95, normalized size = 1.08 \[ -\frac {6 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - {\left (a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{48 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.08, size = 88, normalized size = 1.00 \[ \frac {1}{16} \, a^{2} {\left (\frac {{\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (a x + 1\right )} a^{4}} + \frac {2 \, {\left (a x - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{{\left (a x + 1\right )} a^{4}} + \frac {2 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.81, size = 1695, normalized size = 19.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 268, normalized size = 3.05 \[ -\frac {1}{4} \, {\left (\frac {2 \, x}{a^{2} x^{2} - 1} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \, {\left (2 \, a^{2} x^{2} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{48 \, {\left (a^{5} x^{2} - a^{3}\right )}} - \frac {{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname {artanh}\left (a x\right )}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 213, normalized size = 2.42 \[ \frac {{\ln \left (a\,x+1\right )}^3}{48\,a}-\frac {\ln \left (a\,x+1\right )}{4\,\left (a-a^3\,x^2\right )}-\frac {{\ln \left (1-a\,x\right )}^3}{48\,a}-\frac {x}{4\,a^2\,x^2-4}+\frac {\ln \left (1-a\,x\right )}{4\,a-4\,a^3\,x^2}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{16\,a}-\frac {{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{16\,a}-\frac {x\,{\ln \left (a\,x+1\right )}^2}{8\,\left (a^2\,x^2-1\right )}-\frac {x\,{\ln \left (1-a\,x\right )}^2}{2\,\left (4\,a^2\,x^2-4\right )}+\frac {x\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,a^2\,x^2-4}-\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a} \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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