Optimal. Leaf size=156 \[ \frac {a^2 x}{3}-\frac {1}{3} a \tanh ^{-1}(a x)+\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac {10}{3} a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {5}{3} a \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-a \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.30, antiderivative size = 156, normalized size of antiderivative = 1.00, number
of steps used = 20, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules
used = {6159, 6021, 6131, 6055, 2449, 2352, 6037, 6135, 6079, 2497, 6127, 327, 212}
\begin {gather*} \frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac {a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {5}{3} a \text {Li}_2\left (1-\frac {2}{1-a x}\right )-a \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {1}{3} a \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{x}+\frac {10}{3} a \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 6021
Rule 6037
Rule 6055
Rule 6079
Rule 6127
Rule 6131
Rule 6135
Rule 6159
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^2} \, dx &=\int \left (-2 a^2 \tanh ^{-1}(a x)^2+\frac {\tanh ^{-1}(a x)^2}{x^2}+a^4 x^2 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^2 \tanh ^{-1}(a x)^2 \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+(2 a) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (4 a^3\right ) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^5\right ) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+(2 a) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (4 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx+\frac {1}{3} \left (2 a^3\right ) \int x \tanh ^{-1}(a x) \, dx-\frac {1}{3} \left (2 a^3\right ) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+4 a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} \left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx-\left (4 a^2\right ) \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-\frac {1}{3} a^4 \int \frac {x^2}{1-a^2 x^2} \, dx\\ &=\frac {a^2 x}{3}+\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac {10}{3} a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+(4 a) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )-\frac {1}{3} a^2 \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{3} \left (2 a^2\right ) \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {a^2 x}{3}-\frac {1}{3} a \tanh ^{-1}(a x)+\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac {10}{3} a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+2 a \text {Li}_2\left (1-\frac {2}{1-a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{3} (2 a) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )\\ &=\frac {a^2 x}{3}-\frac {1}{3} a \tanh ^{-1}(a x)+\frac {1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac {2}{3} a \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac {1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac {10}{3} a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {5}{3} a \text {Li}_2\left (1-\frac {2}{1-a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 149, normalized size = 0.96 \begin {gather*} \frac {a^2 x^2-a x \tanh ^{-1}(a x)+a^3 x^3 \tanh ^{-1}(a x)-3 \tanh ^{-1}(a x)^2+8 a x \tanh ^{-1}(a x)^2-6 a^2 x^2 \tanh ^{-1}(a x)^2+a^4 x^4 \tanh ^{-1}(a x)^2+6 a x \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+10 a x \tanh ^{-1}(a x) \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )-5 a x \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-3 a x \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 203, normalized size = 1.30
method | result | size |
derivativedivides | \(a \left (\frac {\arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}-2 \arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right )^{2}}{a x}+\frac {a^{2} x^{2} \arctanh \left (a x \right )}{3}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}+2 \arctanh \left (a x \right ) \ln \left (a x \right )-\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {a x}{3}+\frac {\ln \left (a x -1\right )}{6}-\frac {\ln \left (a x +1\right )}{6}-\dilog \left (a x \right )-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {2 \ln \left (a x -1\right )^{2}}{3}+\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (a x +1\right )^{2}}{3}\right )\) | \(203\) |
default | \(a \left (\frac {\arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}-2 \arctanh \left (a x \right )^{2} a x -\frac {\arctanh \left (a x \right )^{2}}{a x}+\frac {a^{2} x^{2} \arctanh \left (a x \right )}{3}-\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}+2 \arctanh \left (a x \right ) \ln \left (a x \right )-\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {a x}{3}+\frac {\ln \left (a x -1\right )}{6}-\frac {\ln \left (a x +1\right )}{6}-\dilog \left (a x \right )-\dilog \left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {2 \ln \left (a x -1\right )^{2}}{3}+\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {2 \ln \left (a x +1\right )^{2}}{3}\right )\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 200, normalized size = 1.28 \begin {gather*} \frac {1}{6} \, a^{2} {\left (\frac {2 \, {\left (a x + 2 \, \log \left (a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, \log \left (a x - 1\right )^{2}\right )}}{a} + \frac {16 \, {\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 {6 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {6 \, {\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}{3} \, {\left (a^{2} x^{2} - 8 \, \log \left (a x + 1\right ) - 8 \, \log \left (a x - 1\right ) + 6 \, \log \left (x\right )\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{3} \, {\left (a^{4} x^{3} - 6 \, a^{2} x - \frac {3}{x}\right )} \operatorname {artanh}\left (a x\right )^{2} \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 {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{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 )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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