Optimal. Leaf size=91 \[ -\frac {a x}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6177, 6135,
6079, 2497, 6141, 205, 212} \begin {gather*} -\frac {a x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{4} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 2497
Rule 6079
Rule 6135
Rule 6141
Rule 6177
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{2} a \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{4} a \int \frac {1}{1-a^2 x^2} \, dx-a \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a x}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 63, normalized size = 0.69 \begin {gather*} \frac {1}{8} \left (4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x) \left (\cosh \left (2 \tanh ^{-1}(a x)\right )+4 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-4 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-\sinh \left (2 \tanh ^{-1}(a x)\right )\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. \(189\) vs.
\(2(81)=162\).
time = 0.43, size = 190, normalized size = 2.09
method | result | size |
derivativedivides | \(\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\arctanh \left (a x \right )}{4 a x +4}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}\) | \(190\) |
default | \(\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\arctanh \left (a x \right )}{4 a x +4}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}\) | \(190\) |
risch | \(-\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\dilog \left (a x +1\right )}{2}+\frac {\ln \left (a x -1\right )}{16}-\frac {\ln \left (a x +1\right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\ln \left (a x +1\right )}{8 a x +8}+\frac {1}{8 a x +8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (-a x +1\right )^{2}}{8}+\frac {\dilog \left (-a x +1\right )}{2}-\frac {\ln \left (-a x -1\right )}{16}+\frac {\left (-a x +1\right ) \ln \left (-a x +1\right )}{-16 a x -16}-\frac {\ln \left (-a x +1\right )}{8 \left (-a x +1\right )}-\frac {1}{8 \left (-a x +1\right )}+\frac {\left (\ln \left (-a x +1\right )-\ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\dilog \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\) | \(220\) |
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. 206 vs.
\(2 (78) = 156\).
time = 0.27, size = 206, normalized size = 2.26 \begin {gather*} \frac {1}{8} \, a {\left (\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} + 2 \, a x}{a^{3} x^{2} - a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{2} x^{2} - 1} + \log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x \right )}}{x \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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