Optimal. Leaf size=57 \[ -\frac {1}{4 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6145, 6095}
\begin {gather*} -\frac {\tanh ^{-1}(a x)^2}{4 a^3}+\frac {x \tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {1}{4 a^3 \left (1-a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6095
Rule 6145
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx &=-\frac {1}{4 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{2 a^2}\\ &=-\frac {1}{4 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{4 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.79 \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^3 \left (-1+a^2 x^2\right )} \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. \(133\) vs.
\(2(51)=102\).
time = 0.73, size = 134, normalized size = 2.35
method | result | size |
risch | \(-\frac {\ln \left (a x +1\right )^{2}}{16 a^{3}}+\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}-2 a x -\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{8 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{2}-4 a x \ln \left (-a x +1\right )-\ln \left (-a x +1\right )^{2}-4}{16 a^{3} \left (a x -1\right ) \left (a x +1\right )}\) | \(124\) |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )}{4 \left (a x +1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {1}{8 a x -8}-\frac {1}{8 \left (a x +1\right )}}{a^{3}}\) | \(134\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )}{4 \left (a x +1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{16}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {1}{8 a x -8}-\frac {1}{8 \left (a x +1\right )}}{a^{3}}\) | \(134\) |
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. 126 vs.
\(2 (49) = 98\).
time = 0.25, size = 126, normalized size = 2.21 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, x}{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 ({\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}{16 \, {\left (a^{6} x^{2} - a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 65, normalized size = 1.14 \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^{5} x^{2} - a^{3}\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^{2} \operatorname {atanh}{\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 110, normalized size = 1.93 \begin {gather*} \ln \left (1-a\,x\right )\,\left (\frac {\ln \left (a\,x+1\right )}{8\,a^3}+\frac {x}{2\,a^2\,\left (2\,a^2\,x^2-2\right )}\right )-\frac {{\ln \left (a\,x+1\right )}^2}{16\,a^3}-\frac {{\ln \left (1-a\,x\right )}^2}{16\,a^3}-\frac {1}{2\,a^2\,\left (2\,a-2\,a^3\,x^2\right )}-\frac {x\,\ln \left (a\,x+1\right )}{4\,a^3\,\left (a\,x^2-\frac {1}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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