Optimal. Leaf size=138 \[ -\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a^3}+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}-\frac {\tanh ^{-1}(a x)}{30 a^3}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{15 a^3}-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {x}{30 a^2}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {x^3}{30} \]
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Rubi [A] time = 0.41, 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 = {6014, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 302} \[ -\frac {2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{15 a^3}-\frac {1}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {x}{30 a^2}+\frac {2 \tanh ^{-1}(a x)^2}{15 a^3}-\frac {\tanh ^{-1}(a x)}{30 a^3}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{15 a^3}-\frac {1}{10} a x^4 \tanh ^{-1}(a x)+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2+\frac {2 x^2 \tanh ^{-1}(a x)}{15 a}-\frac {x^3}{30} \]
Antiderivative was successfully verified.
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Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rule 6014
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 \operatorname {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 \operatorname {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 = 0.25, size = 95, normalized size = 0.69 \[ -\frac {a^3 x^3+2 \left (3 a^5 x^5-5 a^3 x^3+2\right ) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (3 a^4 x^4-4 a^2 x^2+8 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+1\right )-4 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-a x}{30 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} x^{4} - x^{2}\right )} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
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 \[ \int -{\left (a^{2} x^{2} - 1\right )} x^{2} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 205, normalized size = 1.49 \[ -\frac {a^{2} x^{5} \arctanh \left (a x \right )^{2}}{5}+\frac {x^{3} \arctanh \left (a x \right )^{2}}{3}-\frac {a \,x^{4} \arctanh \left (a x \right )}{10}+\frac {2 x^{2} \arctanh \left (a x \right )}{15 a}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{15 a^{3}}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{15 a^{3}}+\frac {\ln \left (a x -1\right )^{2}}{30 a^{3}}-\frac {2 \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{15 a^{3}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{15 a^{3}}-\frac {\ln \left (a x +1\right )^{2}}{30 a^{3}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{15 a^{3}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{15 a^{3}}-\frac {x^{3}}{30}+\frac {x}{30 a^{2}}+\frac {\ln \left (a x -1\right )}{60 a^{3}}-\frac {\ln \left (a x +1\right )}{60 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 173, normalized size = 1.25 \[ -\frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} - 2 \, a x + 2 \, \log \left (a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )}{a^{5}} + \frac {8 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {\log \left (a x + 1\right )}{a^{5}}\right )} - \frac {1}{30} \, a {\left (\frac {3 \, a^{2} x^{4} - 4 \, x^{2}}{a^{2}} - \frac {4 \, \log \left (a x + 1\right )}{a^{4}} - \frac {4 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {1}{15} \, {\left (3 \, a^{2} x^{5} - 5 \, x^{3}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
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 \[ -\int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right ) \,d x \]
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 \[ - \int \left (- x^{2} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{2} x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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