Optimal. Leaf size=62 \[ \frac {1}{2} a^4 x^2 \tanh ^{-1}(a x)+\frac {a^3 x}{2}+a^2 \text {Li}_2(-a x)-a^2 \text {Li}_2(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
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Rubi [A] time = 0.09, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6012, 5916, 325, 206, 5912, 321} \[ a^2 \text {PolyLog}(2,-a x)-a^2 \text {PolyLog}(2,a x)+\frac {1}{2} a^4 x^2 \tanh ^{-1}(a x)+\frac {a^3 x}{2}-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 325
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^3} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)}{x^3}-\frac {2 a^2 \tanh ^{-1}(a x)}{x}+a^4 x \tanh ^{-1}(a x)\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x} \, dx\right )+a^4 \int x \tanh ^{-1}(a x) \, dx+\int \frac {\tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^4 x^2 \tanh ^{-1}(a x)+a^2 \text {Li}_2(-a x)-a^2 \text {Li}_2(a x)+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx-\frac {1}{2} a^5 \int \frac {x^2}{1-a^2 x^2} \, dx\\ &=-\frac {a}{2 x}+\frac {a^3 x}{2}-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^4 x^2 \tanh ^{-1}(a x)+a^2 \text {Li}_2(-a x)-a^2 \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 1.00 \[ -\frac {-a^4 x^4 \tanh ^{-1}(a x)-a^3 x^3-2 a^2 x^2 \text {Li}_2(-a x)+2 a^2 x^2 \text {Li}_2(a x)+a x+\tanh ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 1.29 \[ \frac {a^{4} x^{2} \arctanh \left (a x \right )}{2}-2 a^{2} \arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{2 x^{2}}+a^{2} \dilog \left (a x \right )+a^{2} \dilog \left (a x +1\right )+a^{2} \ln \left (a x \right ) \ln \left (a x +1\right )+\frac {a^{3} x}{2}-\frac {a}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 82, normalized size = 1.32 \[ \frac {1}{2} \, {\left (2 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a - 2 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a + \frac {a^{2} x^{2} - 1}{x}\right )} a + \frac {1}{2} \, {\left (a^{4} x^{2} - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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