Optimal. Leaf size=125 \[ \frac {3}{32 a^2 \left (1-a^2 x^2\right )}+\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^2} \]
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Rubi [A] time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5994, 5960, 5956, 261} \[ \frac {3}{32 a^2 \left (1-a^2 x^2\right )}+\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5956
Rule 5960
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\tanh ^{-1}(a x)^2}{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 )^3} \, dx}{2 a}\\ &=\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{8 a}\\ &=\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^2}+\frac {\tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {3}{16} \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {1}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {3}{32 a^2 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{8 a \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{16 a \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{32 a^2}+\frac {\tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 71, normalized size = 0.57 \[ \frac {-3 a^2 x^2+2 a x \left (3 a^2 x^2-5\right ) \tanh ^{-1}(a x)+\left (-3 a^4 x^4+6 a^2 x^2+5\right ) \tanh ^{-1}(a x)^2+4}{32 a^2 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 99, normalized size = 0.79 \[ -\frac {12 \, a^{2} x^{2} + {\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{128 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 251, normalized size = 2.01 \[ -\frac {1}{512} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {4 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} + \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} - \frac {8 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {16 \, {\left (a x + 1\right )}}{a x - 1} - 1\right )}}{{\left (a x + 1\right )}^{2} a^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2} a^{3}} + \frac {16 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 247, normalized size = 1.98 \[ \frac {\arctanh \left (a x \right )^{2}}{4 a^{2} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\arctanh \left (a x \right )}{32 a^{2} \left (a x -1\right )^{2}}+\frac {3 \arctanh \left (a x \right )}{32 a^{2} \left (a x -1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{32 a^{2}}+\frac {\arctanh \left (a x \right )}{32 a^{2} \left (a x +1\right )^{2}}+\frac {3 \arctanh \left (a x \right )}{32 a^{2} \left (a x +1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{32 a^{2}}+\frac {3 \ln \left (a x -1\right )^{2}}{128 a^{2}}-\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{64 a^{2}}+\frac {3 \ln \left (a x +1\right )^{2}}{128 a^{2}}+\frac {3 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{64 a^{2}}-\frac {3 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{64 a^{2}}+\frac {1}{128 a^{2} \left (a x -1\right )^{2}}-\frac {7}{128 a^{2} \left (a x -1\right )}+\frac {1}{128 a^{2} \left (a x +1\right )^{2}}+\frac {7}{128 a^{2} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 206, normalized size = 1.65 \[ \frac {{\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )}{32 \, a} - \frac {12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16}{128 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} + \frac {\operatorname {artanh}\left (a x\right )^{2}}{4 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 319, normalized size = 2.55 \[ {\ln \left (a\,x+1\right )}^2\,\left (\frac {1}{16\,a^3\,\left (\frac {1}{a}-2\,a\,x^2+a^3\,x^4\right )}-\frac {3}{128\,a^2}\right )-{\ln \left (1-a\,x\right )}^2\,\left (\frac {3}{128\,a^2}-\frac {1}{4\,a^2\,\left (4\,a^4\,x^4-8\,a^2\,x^2+4\right )}\right )-\ln \left (1-a\,x\right )\,\left (\frac {\frac {1}{4\,a}-\frac {5\,x}{8}+\frac {3\,a^2\,x^3}{8}}{8\,a^5\,x^4-16\,a^3\,x^2+8\,a}-\frac {\frac {5\,x}{8}+\frac {1}{4\,a}-\frac {3\,a^2\,x^3}{8}}{8\,a^5\,x^4-16\,a^3\,x^2+8\,a}+\ln \left (a\,x+1\right )\,\left (\frac {1}{4\,a^2\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}-\frac {3\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}{32\,a^2\,\left (2\,a^4\,x^4-4\,a^2\,x^2+2\right )}\right )\right )+\frac {\frac {2}{a^2}-\frac {3\,x^2}{2}}{16\,a^4\,x^4-32\,a^2\,x^2+16}-\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{32\,a^2}-\frac {3\,x^3}{32}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \]
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 \[ - \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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