Optimal. Leaf size=138 \[ \frac {x}{30 a^2}-\frac {x^3}{30}-\frac {\tanh ^{-1}(a x)}{30 a^3}+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{15 a^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6161, 6037,
6127, 327, 212, 6131, 6055, 2449, 2352, 308} \begin {gather*} -\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a^3}+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}-\frac {\tanh ^{-1}(a x)}{30 a^3}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{15 a^3}-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {x}{30 a^2}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {x^3}{30} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rule 6161
Rubi steps
\begin {align*} \int x^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int x^4 \tanh ^{-1}(a x)^2 \, dx\right )+\int x^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {1}{3} (2 a) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{5} \left (2 a^3\right ) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{3 a}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}-\frac {1}{5} (2 a) \int x^3 \tanh ^{-1}(a x) \, dx+\frac {1}{5} (2 a) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \tanh ^{-1}(a x)}{3 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {1}{3} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{3 a^2}-\frac {2 \int x \tanh ^{-1}(a x) \, dx}{5 a}+\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}+\frac {1}{10} a^2 \int \frac {x^4}{1-a^2 x^2} \, dx\\ &=\frac {x}{3 a^2}+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}+\frac {1}{5} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{3 a^2}+\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^2}+\frac {1}{10} a^2 \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {x}{30 a^2}-\frac {x^3}{30}-\frac {\tanh ^{-1}(a x)}{3 a^3}+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{3 a^3}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^2}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^2}-\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^2}\\ &=\frac {x}{30 a^2}-\frac {x^3}{30}-\frac {\tanh ^{-1}(a x)}{30 a^3}+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{3 a^3}+\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^3}\\ &=\frac {x}{30 a^2}-\frac {x^3}{30}-\frac {\tanh ^{-1}(a x)}{30 a^3}+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 95, normalized size = 0.69 \begin {gather*} -\frac {-a x+a^3 x^3+2 \left (2-5 a^3 x^3+3 a^5 x^5\right ) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (1-4 a^2 x^2+3 a^4 x^4+8 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )\right )-4 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )}{30 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 179, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {a^{4} x^{4} \arctanh \left (a x \right )}{10}+\frac {2 a^{2} x^{2} \arctanh \left (a x \right )}{15}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}-\frac {a^{3} x^{3}}{30}+\frac {a x}{30}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}}{a^{3}}\) | \(179\) |
default | \(\frac {-\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {\arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {a^{4} x^{4} \arctanh \left (a x \right )}{10}+\frac {2 a^{2} x^{2} \arctanh \left (a x \right )}{15}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {\ln \left (a x -1\right )^{2}}{30}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {\ln \left (a x +1\right )^{2}}{30}-\frac {a^{3} x^{3}}{30}+\frac {a x}{30}+\frac {\ln \left (a x -1\right )}{60}-\frac {\ln \left (a x +1\right )}{60}}{a^{3}}\) | \(179\) |
risch | \(-\frac {443}{3375 a^{3}}-\frac {x^{3}}{30}-\frac {\left (\left (-\frac {1}{9}+\frac {\ln \left (a x +1\right )}{3}\right ) \left (a x +1\right )^{3}+\left (\frac {1}{2}-\ln \left (a x +1\right )\right ) \left (a x +1\right )^{2}+\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right )\right ) \ln \left (-a x +1\right )}{2 a^{3}}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{6 a^{3}}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{3 a^{3}}-\frac {\left (a x +1\right )^{2} \ln \left (a x +1\right )}{12 a^{3}}+\frac {\left (a x +1\right )^{3} \ln \left (a x +1\right )}{18 a^{3}}+\frac {\ln \left (-a x +1\right ) \ln \left (a x +1\right )}{10 a^{3}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{5 a^{3}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{3}}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}-\frac {31 \ln \left (a x -1\right )}{225 a^{3}}+\frac {\ln \left (a x +1\right )^{2}}{30 a^{3}}+\frac {\ln \left (a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (a x +1\right ) x^{3}}{18}-\frac {7 \ln \left (a x +1\right )}{45 a^{3}}-\frac {\ln \left (-a x +1\right )^{2}}{30 a^{3}}-\frac {34 \ln \left (-a x +1\right )}{225 a^{3}}+\frac {\ln \left (-a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (-a x +1\right ) x^{3}}{18}+\frac {x}{30 a^{2}}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{5}}{20}-\frac {a \ln \left (a x +1\right ) x^{4}}{20}-\frac {\ln \left (a x +1\right ) x^{2}}{60 a}-\frac {\ln \left (a x +1\right ) x}{6 a^{2}}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{5}}{20}+\frac {a \ln \left (-a x +1\right ) x^{4}}{20}+\frac {\ln \left (-a x +1\right ) x^{2}}{60 a}-\frac {\ln \left (-a x +1\right ) x}{6 a^{2}}-\frac {2 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a^{3}}\) | \(472\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 173, normalized size = 1.25 \begin {gather*} -\frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} - 2 \, 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} - \log \left (a x - 1\right )}{a^{5}} + \frac {8 \, {\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^{5}} + \frac {\log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{30} \, a {\left (\frac {3 \, a^{2} x^{4} - 4 \, x^{2}}{a^{2}} - \frac {4 \, \log \left (a x + 1\right )}{a^{4}} - \frac {4 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {1}{15} \, {\left (3 \, a^{2} x^{5} - 5 \, x^{3}\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 \left (- x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\, 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 x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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