Optimal. Leaf size=171 \[ -\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8 \tanh ^{-1}(a x)^2}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a}-\frac {8 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{15 a} \]
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Rubi [A]
time = 0.10, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6091, 6021,
6131, 6055, 2449, 2352, 8} \begin {gather*} \frac {a^2 x^3}{30}+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {8 \tanh ^{-1}(a x)^2}{15 a}-\frac {16 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{15 a}-\frac {11 x}{30} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6091
Rule 6131
Rubi steps
\begin {align*} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {1}{10} \int \left (1-a^2 x^2\right ) \, dx+\frac {4}{5} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {x}{10}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {4 \int 1 \, dx}{15}+\frac {8}{15} \int \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {1}{15} (16 a) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8 \tanh ^{-1}(a x)^2}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {16}{15} \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8 \tanh ^{-1}(a x)^2}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a}+\frac {16}{15} \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8 \tanh ^{-1}(a x)^2}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a}-\frac {16 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{15 a}\\ &=-\frac {11 x}{30}+\frac {a^2 x^3}{30}+\frac {4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{15 a}+\frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{10 a}+\frac {8 \tanh ^{-1}(a x)^2}{15 a}+\frac {8}{15} x \tanh ^{-1}(a x)^2+\frac {4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 99, normalized size = 0.58 \begin {gather*} \frac {a x \left (-11+a^2 x^2\right )+2 (-1+a x)^3 \left (8+9 a x+3 a^2 x^2\right ) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (11-14 a^2 x^2+3 a^4 x^4-32 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )\right )+16 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )}{30 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 188, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {2 \arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}+\arctanh \left (a x \right )^{2} a x +\frac {a^{4} x^{4} \arctanh \left (a x \right )}{10}-\frac {7 a^{2} x^{2} \arctanh \left (a x \right )}{15}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{3} x^{3}}{30}-\frac {11 a x}{30}-\frac {11 \ln \left (a x -1\right )}{60}+\frac {11 \ln \left (a x +1\right )}{60}+\frac {2 \ln \left (a x -1\right )^{2}}{15}-\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {2 \ln \left (a x +1\right )^{2}}{15}}{a}\) | \(188\) |
default | \(\frac {\frac {\arctanh \left (a x \right )^{2} a^{5} x^{5}}{5}-\frac {2 \arctanh \left (a x \right )^{2} a^{3} x^{3}}{3}+\arctanh \left (a x \right )^{2} a x +\frac {a^{4} x^{4} \arctanh \left (a x \right )}{10}-\frac {7 a^{2} x^{2} \arctanh \left (a x \right )}{15}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15}+\frac {a^{3} x^{3}}{30}-\frac {11 a x}{30}-\frac {11 \ln \left (a x -1\right )}{60}+\frac {11 \ln \left (a x +1\right )}{60}+\frac {2 \ln \left (a x -1\right )^{2}}{15}-\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15}+\frac {4 \left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{15}-\frac {2 \ln \left (a x +1\right )^{2}}{15}}{a}\) | \(188\) |
risch | \(-\frac {11 x}{30}-\frac {94 \ln \left (a x -1\right )}{225 a}-\frac {a^{4} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{5}}{10}+\frac {a^{2} x^{3}}{30}+\frac {a^{2} \ln \left (-a x +1\right ) \ln \left (a x +1\right ) x^{3}}{3}-\frac {\left (-1+\ln \left (a x +1\right )\right ) \left (a x +1\right ) \ln \left (-a x +1\right )}{2 a}-\frac {x \ln \left (-a x +1\right )}{2}-\frac {a^{2} \ln \left (a x +1\right )^{2} x^{3}}{6}-\frac {7 a \ln \left (a x +1\right ) x^{2}}{30}-\frac {3739}{6750 a}+\frac {\ln \left (-a x +1\right )^{2} x}{4}-\frac {2 \ln \left (-a x +1\right )^{2}}{15 a}-\frac {239 \ln \left (-a x +1\right )}{900 a}+\frac {\ln \left (a x +1\right )^{2} x}{4}+\frac {2 \ln \left (a x +1\right )^{2}}{15 a}-\frac {\ln \left (a x +1\right ) x}{2}-\frac {19 \ln \left (a x +1\right )}{60 a}+\frac {a^{4} \ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {a^{3} \ln \left (a x +1\right ) x^{4}}{20}-\frac {a^{2} \ln \left (-a x +1\right )^{2} x^{3}}{6}+\frac {7 a \ln \left (-a x +1\right ) x^{2}}{30}-\frac {a^{3} \ln \left (-a x +1\right ) x^{4}}{20}+\frac {a^{4} \ln \left (-a x +1\right )^{2} x^{5}}{20}+\frac {7 \ln \left (-a x +1\right ) \ln \left (a x +1\right )}{30 a}-\frac {7 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{15 a}+\frac {7 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a}+\frac {\left (a x +1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {8 \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{15 a}\) | \(418\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 175, normalized size = 1.02 \begin {gather*} \frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} - 22 \, a x - 8 \, \log \left (a x + 1\right )^{2} + 16 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 8 \, \log \left (a x - 1\right )^{2} - 11 \, \log \left (a x - 1\right )}{a^{3}} - \frac {32 \, {\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^{3}} + \frac {11 \, \log \left (a x + 1\right )}{a^{3}}\right )} + \frac {1}{30} \, {\left (3 \, a^{2} x^{4} - 14 \, x^{2} + \frac {16 \, \log \left (a x + 1\right )}{a^{2}} + \frac {16 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {1}{15} \, {\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, 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 \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \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 {\mathrm {atanh}\left (a\,x\right )}^2\,{\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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