Optimal. Leaf size=186 \[ \frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {a^2 x^2}{12}-\frac {2}{3} \log \left (1-a^2 x^2\right )-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (\frac {2}{1-a x}-1\right )-\text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+\text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right ) \]
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Rubi [A] time = 0.53, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6012, 5914, 6052, 5948, 6058, 6610, 5916, 5980, 5910, 260, 266, 43} \[ \frac {1}{2} \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {PolyLog}\left (3,\frac {2}{1-a x}-1\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{1-a x}-1\right )+\frac {a^2 x^2}{12}-\frac {2}{3} \log \left (1-a^2 x^2\right )+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)^2-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 5910
Rule 5914
Rule 5916
Rule 5948
Rule 5980
Rule 6012
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+a^4 x^3 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^3 \tanh ^{-1}(a x)^2 \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-(4 a) \int \frac {\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^3\right ) \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{2} a^5 \int \frac {x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-(2 a) \int \tanh ^{-1}(a x) \, dx+(2 a) \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{2} a^3 \int x^2 \tanh ^{-1}(a x) \, dx-\frac {1}{2} a^3 \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-2 a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a \int \tanh ^{-1}(a x) \, dx-\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+a \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-a \int \frac {\text {Li}_2\left (-1+\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^2\right ) \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{6} a^4 \int \frac {x^3}{1-a^2 x^2} \, dx\\ &=-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )-\frac {1}{2} a^2 \int \frac {x}{1-a^2 x^2} \, dx-\frac {1}{12} a^4 \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {3}{4} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )-\frac {1}{12} a^4 \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{12}-\frac {3}{2} a x \tanh ^{-1}(a x)+\frac {1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac {3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {2}{3} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1-a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 191, normalized size = 1.03 \[ \frac {1}{4} \left (a^4 x^4-1\right ) \tanh ^{-1}(a x)^2+\frac {a^2 x^2}{12}-\frac {2}{3} \log \left (1-a^2 x^2\right )+\frac {1}{6} a x \left (a^2 x^2+3\right ) \tanh ^{-1}(a x)-\left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2-\frac {1}{2} \text {Li}_3\left (\frac {-a x-1}{a x-1}\right )+\frac {1}{2} \text {Li}_3\left (\frac {a x+1}{a x-1}\right )+\text {Li}_2\left (\frac {-a x-1}{a x-1}\right ) \tanh ^{-1}(a x)-\text {Li}_2\left (\frac {a x+1}{a x-1}\right ) \tanh ^{-1}(a x)-2 a x \tanh ^{-1}(a x)+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.80, size = 728, normalized size = 3.91 \[ -a^{2} x^{2} \arctanh \left (a x \right )^{2}-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2}}{2}-\frac {1}{12}+\frac {3 \arctanh \left (a x \right )^{2}}{4}+\frac {a^{4} x^{4} \arctanh \left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{12}-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2}}{2}+\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )}{2}-2 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3}}{2}+\arctanh \left (a x \right )^{2} \ln \left (a x \right )-\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\left (a x +1\right ) \arctanh \left (a x \right )-2 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {4 \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{3}+\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {\left (a^{2} x^{2}-4 a x +7\right ) \left (a x +1\right ) \arctanh \left (a x \right )}{6}+\frac {\left (a x -3\right ) \left (a x +1\right ) \arctanh \left (a x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, {\left (a^{4} x^{4} - 4 \, a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{2} - \frac {1}{4} \, \int -\frac {2 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )^{2} - {\left (a^{5} x^{5} - 4 \, a^{3} x^{3} + 4 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \, {\left (a x^{2} - x\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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