Optimal. Leaf size=70 \[ \frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)+\frac {a^3 x^3}{12}-a^2 x^2 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}-\frac {3 a x}{4}+\frac {3}{4} \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.10, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6012, 5912, 5916, 321, 206, 302} \[ -\frac {1}{2} \text {PolyLog}(2,-a x)+\frac {1}{2} \text {PolyLog}(2,a x)+\frac {a^3 x^3}{12}+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)-\frac {3 a x}{4}+\frac {3}{4} \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 302
Rule 321
Rule 5912
Rule 5916
Rule 6012
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{x} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)}{x}-2 a^2 x \tanh ^{-1}(a x)+a^4 x^3 \tanh ^{-1}(a x)\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x \tanh ^{-1}(a x) \, dx\right )+a^4 \int x^3 \tanh ^{-1}(a x) \, dx+\int \frac {\tanh ^{-1}(a x)}{x} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}+a^3 \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {1}{4} a^5 \int \frac {x^4}{1-a^2 x^2} \, dx\\ &=-a x-a^2 x^2 \tanh ^{-1}(a x)+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}+a \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{4} a^5 \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 {3 a x}{4}+\frac {a^3 x^3}{12}+\tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}-\frac {1}{4} a \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {3 a x}{4}+\frac {a^3 x^3}{12}+\frac {3}{4} \tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)+\frac {1}{4} a^4 x^4 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 73, normalized size = 1.04 \[ \frac {1}{24} \left (6 a^4 x^4 \tanh ^{-1}(a x)+2 a^3 x^3-24 a^2 x^2 \tanh ^{-1}(a x)-12 \text {Li}_2(-a x)+12 \text {Li}_2(a x)-18 a x-9 \log (1-a x)+9 \log (a x+1)\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 1.27 \[ \frac {a^{4} x^{4} \arctanh \left (a x \right )}{4}-a^{2} x^{2} \arctanh \left (a x \right )+\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}+\frac {x^{3} a^{3}}{12}-\frac {3 a x}{4}-\frac {3 \ln \left (a x -1\right )}{8}+\frac {3 \ln \left (a x +1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 106, normalized size = 1.51 \[ \frac {1}{24} \, {\left (2 \, a^{2} x^{3} - 18 \, x - \frac {12 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {12 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a} + \frac {9 \, \log \left (a x + 1\right )}{a} - \frac {9 \, \log \left (a x - 1\right )}{a}\right )} a + \frac {1}{4} \, {\left (a^{4} x^{4} - 4 \, a^{2} x^{2} + 2 \, \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x} \,d x \]
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 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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