Optimal. Leaf size=77 \[ -\frac {1}{2} a^4 \text {Li}_2(-a x)+\frac {1}{2} a^4 \text {Li}_2(a x)-\frac {3}{4} a^4 \tanh ^{-1}(a x)+\frac {3 a^3}{4 x}+\frac {a^2 \tanh ^{-1}(a x)}{x^2}-\frac {\tanh ^{-1}(a x)}{4 x^4}-\frac {a}{12 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6012, 5916, 325, 206, 5912} \[ -\frac {1}{2} a^4 \text {PolyLog}(2,-a x)+\frac {1}{2} a^4 \text {PolyLog}(2,a x)+\frac {a^2 \tanh ^{-1}(a x)}{x^2}+\frac {3 a^3}{4 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)-\frac {a}{12 x^3}-\frac {\tanh ^{-1}(a x)}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 206
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^5} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)}{x^5}-\frac {2 a^2 \tanh ^{-1}(a x)}{x^3}+\frac {a^4 \tanh ^{-1}(a x)}{x}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)}{x} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^5} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)}{x^2}-\frac {1}{2} a^4 \text {Li}_2(-a x)+\frac {1}{2} a^4 \text {Li}_2(a x)+\frac {1}{4} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx-a^3 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{12 x^3}+\frac {a^3}{x}-\frac {\tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)}{x^2}-\frac {1}{2} a^4 \text {Li}_2(-a x)+\frac {1}{2} a^4 \text {Li}_2(a x)+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx-a^5 \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a}{12 x^3}+\frac {3 a^3}{4 x}-a^4 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)}{x^2}-\frac {1}{2} a^4 \text {Li}_2(-a x)+\frac {1}{2} a^4 \text {Li}_2(a x)+\frac {1}{4} a^5 \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a}{12 x^3}+\frac {3 a^3}{4 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)}{x^2}-\frac {1}{2} a^4 \text {Li}_2(-a x)+\frac {1}{2} a^4 \text {Li}_2(a x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 1.09 \[ \frac {1}{24} \left (-12 a^4 \text {Li}_2(-a x)+12 a^4 \text {Li}_2(a x)+9 a^4 \log (1-a x)-9 a^4 \log (a x+1)+\frac {18 a^3}{x}+\frac {24 a^2 \tanh ^{-1}(a x)}{x^2}-\frac {6 \tanh ^{-1}(a x)}{x^4}-\frac {2 a}{x^3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 105, normalized size = 1.36 \[ a^{4} \arctanh \left (a x \right ) \ln \left (a x \right )+\frac {a^{2} \arctanh \left (a x \right )}{x^{2}}-\frac {\arctanh \left (a x \right )}{4 x^{4}}-\frac {a^{4} \dilog \left (a x \right )}{2}-\frac {a^{4} \dilog \left (a x +1\right )}{2}-\frac {a^{4} \ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {a}{12 x^{3}}+\frac {3 a^{3}}{4 x}+\frac {3 a^{4} \ln \left (a x -1\right )}{8}-\frac {3 a^{4} \ln \left (a x +1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 112, normalized size = 1.45 \[ -\frac {1}{24} \, {\left (12 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a^{3} - 12 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a^{3} + 9 \, a^{3} \log \left (a x + 1\right ) - 9 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (9 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a + \frac {1}{4} \, {\left (2 \, a^{4} \log \left (x^{2}\right ) + \frac {4 \, a^{2} x^{2} - 1}{x^{4}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2}{x^5} \,d x \]
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 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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