Optimal. Leaf size=54 \[ -\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6103, 267}
\begin {gather*} -\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6103
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 a}-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.81 \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 \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.
\(2(48)=96\).
time = 0.73, size = 134, normalized size = 2.48
method | result | size |
risch | \(\frac {\ln \left (a x +1\right )^{2}}{16 a}-\frac {\left (x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x -\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{8 \left (a^{2} x^{2}-1\right ) a}+\frac {a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a x \ln \left (-a x +1\right )-\ln \left (-a x +1\right )^{2}+4}{16 a \left (a x -1\right ) \left (a x +1\right )}\) | \(124\) |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )}{4 \left (a x +1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )^{2}}{16}+\frac {1}{8 a x -8}-\frac {1}{8 \left (a x +1\right )}}{a}\) | \(134\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )}{4 \left (a x +1\right )}+\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{4}-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{4}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x -1\right )^{2}}{16}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (a x +1\right )^{2}}{16}+\frac {1}{8 a x -8}-\frac {1}{8 \left (a x +1\right )}}{a}\) | \(134\) |
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. 122 vs.
\(2 (46) = 92\).
time = 0.26, size = 122, normalized size = 2.26 \begin {gather*} -\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 ) - \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^{4} x^{2} - a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 64, normalized size = 1.19 \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^{3} x^{2} - a\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 {\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs.
\(2 (46) = 92\).
time = 1.26, size = 255, normalized size = 4.72 \begin {gather*} \frac {1}{8} \, a^{2} {\left (\frac {{\left (a x - 1\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} - 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{a - \frac {a {\left (\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1\right )}}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}} + 1}\right )}{{\left (a x + 1\right )} a^{4}} + \frac {a x - 1}{{\left (a x + 1\right )} a^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.97, size = 106, normalized size = 1.96 \begin {gather*} \frac {{\ln \left (a\,x+1\right )}^2}{16\,a}-\ln \left (1-a\,x\right )\,\left (\frac {\ln \left (a\,x+1\right )}{8\,a}-\frac {x}{2\,\left (2\,a^2\,x^2-2\right )}\right )+\frac {{\ln \left (1-a\,x\right )}^2}{16\,a}+\frac {1}{2\,a\,\left (2\,a^2\,x^2-2\right )}-\frac {x\,\ln \left (a\,x+1\right )}{4\,a\,\left (a\,x^2-\frac {1}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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