Optimal. Leaf size=82 \[ \frac {1}{4 a^2 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 82, 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 = {6141, 6103,
267} \begin {gather*} \frac {1}{4 a^2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6103
Rule 6141
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a}\\ &=-\frac {x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}+\frac {1}{2} \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {1}{4 a^2 \left (1-a^2 x^2\right )}-\frac {x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^2}+\frac {\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.52 \begin {gather*} \frac {1-2 a x \tanh ^{-1}(a x)+\left (1+a^2 x^2\right ) \tanh ^{-1}(a x)^2}{4 a^2-4 a^4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(152\) vs.
\(2(74)=148\).
time = 0.38, size = 153, normalized size = 1.87
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}+1\right ) \ln \left (a x +1\right )^{2}}{16 a^{2} \left (a x -1\right ) \left (a x +1\right )}+\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{8 a^{2} \left (a x -1\right ) \left (a x +1\right )}-\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}+4}{16 a^{2} \left (a x -1\right ) \left (a x +1\right )}\) | \(146\) |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )^{2}}{2 \left (a^{2} x^{2}-1\right )}+\frac {\arctanh \left (a x \right )}{4 a x -4}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\arctanh \left (a x \right )}{4 a x +4}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {1}{8 a x +8}-\frac {1}{8 \left (a x -1\right )}}{a^{2}}\) | \(153\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )^{2}}{2 \left (a^{2} x^{2}-1\right )}+\frac {\arctanh \left (a x \right )}{4 a x -4}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\arctanh \left (a x \right )}{4 a x +4}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {1}{8 a x +8}-\frac {1}{8 \left (a x -1\right )}}{a^{2}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 146 vs.
\(2 (71) = 142\).
time = 0.26, size = 146, normalized size = 1.78 \begin {gather*} \frac {{\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 )}{4 \, a} + \frac {{\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}{16 \, {\left (a^{4} x^{2} - a^{2}\right )}} - \frac {\operatorname {artanh}\left (a x\right )^{2}}{2 \, {\left (a^{2} x^{2} - 1\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 66, normalized size = 0.80 \begin {gather*} \frac {4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4}{16 \, {\left (a^{4} x^{2} - a^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 140, normalized size = 1.71 \begin {gather*} -\frac {1}{32} \, {\left ({\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 2 \, {\left (\frac {a x + 1}{{\left (a x - 1\right )} a^{3}} - \frac {a x - 1}{{\left (a x + 1\right )} a^{3}}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + \frac {2 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{3}} + \frac {2 \, {\left (a x - 1\right )}}{{\left (a x + 1\right )} a^{3}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 198, normalized size = 2.41 \begin {gather*} \ln \left (1-a\,x\right )\,\left (\frac {\frac {x}{2}-\frac {1}{2\,a}}{4\,a-4\,a^3\,x^2}+\frac {\frac {x}{2}+\frac {1}{2\,a}}{4\,a-4\,a^3\,x^2}+\ln \left (a\,x+1\right )\,\left (\frac {1}{8\,a^2}+\frac {1}{2\,a^2\,\left (2\,a^2\,x^2-2\right )}\right )\right )-{\ln \left (1-a\,x\right )}^2\,\left (\frac {1}{16\,a^2}+\frac {1}{2\,a^2\,\left (4\,a^2\,x^2-4\right )}\right )-\frac {1}{2\,a^2\,\left (2\,a^2\,x^2-2\right )}-{\ln \left (a\,x+1\right )}^2\,\left (\frac {1}{8\,a^3\,\left (a\,x^2-\frac {1}{a}\right )}+\frac {1}{16\,a^2}\right )+\frac {x\,\ln \left (a\,x+1\right )}{4\,a^2\,\left (a\,x^2-\frac {1}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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