Optimal. Leaf size=100 \[ -\frac {\tanh ^{-1}(a x)^2}{16 a^3}-\frac {x \tanh ^{-1}(a x)}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {1}{16 a^3 \left (1-a^2 x^2\right )}-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2} \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5998, 5956, 261} \[ \frac {1}{16 a^3 \left (1-a^2 x^2\right )}-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5956
Rule 5998
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {\int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^2}\\ &=-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{16 a^3}+\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{8 a}\\ &=-\frac {1}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {1}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{16 a^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 61, normalized size = 0.61 \[ -\frac {-2 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)+a^2 x^2+\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2}{16 a^3 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 95, normalized size = 0.95 \[ -\frac {4 \, a^{2} x^{2} + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{64 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \]
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 \[ \int -\frac {x^{2} \operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 225, normalized size = 2.25 \[ \frac {\arctanh \left (a x \right )}{16 a^{3} \left (a x -1\right )^{2}}+\frac {\arctanh \left (a x \right )}{16 a^{3} \left (a x -1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{16 a^{3}}-\frac {\arctanh \left (a x \right )}{16 a^{3} \left (a x +1\right )^{2}}+\frac {\arctanh \left (a x \right )}{16 a^{3} \left (a x +1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{16 a^{3}}+\frac {\ln \left (a x +1\right )^{2}}{64 a^{3}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{32 a^{3}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{32 a^{3}}+\frac {\ln \left (a x -1\right )^{2}}{64 a^{3}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{32 a^{3}}-\frac {1}{64 a^{3} \left (a x -1\right )^{2}}-\frac {1}{64 a^{3} \left (a x -1\right )}-\frac {1}{64 a^{3} \left (a x +1\right )^{2}}+\frac {1}{64 a^{3} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 179, normalized size = 1.79 \[ \frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 150, normalized size = 1.50 \[ \ln \left (1-a\,x\right )\,\left (\frac {\ln \left (a\,x+1\right )}{32\,a^3}-\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )-\frac {{\ln \left (a\,x+1\right )}^2}{64\,a^3}-\frac {{\ln \left (1-a\,x\right )}^2}{64\,a^3}-\frac {x^2}{2\,\left (8\,a^5\,x^4-16\,a^3\,x^2+8\,a\right )}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {x}{16\,a^3}+\frac {x^3}{16\,a}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \]
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 \[ - \int \frac {x^{2} \operatorname {atanh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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