Optimal. Leaf size=116 \[ \frac {2}{3} a^3 \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2+\frac {1}{3} a^3 \tanh ^{-1}(a x)-\frac {4}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a^2}{3 x}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-\frac {\tanh ^{-1}(a x)^2}{3 x^3}-\frac {a \tanh ^{-1}(a x)}{3 x^2} \]
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Rubi [A] time = 0.31, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6014, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ \frac {2}{3} a^3 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {a^2}{3 x}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2+\frac {1}{3} a^3 \tanh ^{-1}(a x)+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-\frac {4}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {\tanh ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 2447
Rule 5916
Rule 5932
Rule 5982
Rule 5988
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x^4} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x^4} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}+\frac {1}{3} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}+\frac {1}{3} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-2 a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-\frac {4}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+a^3 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\frac {1}{3} a^4 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}+\frac {1}{3} a^3 \tanh ^{-1}(a x)-\frac {a \tanh ^{-1}(a x)}{3 x^2}-\frac {2}{3} a^3 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{3 x^3}+\frac {a^2 \tanh ^{-1}(a x)^2}{x}-\frac {4}{3} a^3 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {2}{3} a^3 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.31, size = 93, normalized size = 0.80 \[ \frac {2 a^3 x^3 \text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (a^3 x^3-4 a^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-a x\right )-a^2 x^2-(a x-1)^2 (2 a x+1) \tanh ^{-1}(a x)^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{4}}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 237, normalized size = 2.04 \[ \frac {a^{2} \arctanh \left (a x \right )^{2}}{x}-\frac {\arctanh \left (a x \right )^{2}}{3 x^{3}}-\frac {a \arctanh \left (a x \right )}{3 x^{2}}-\frac {4 a^{3} \arctanh \left (a x \right ) \ln \left (a x \right )}{3}+\frac {2 a^{3} \arctanh \left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {2 a^{3} \arctanh \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {a^{2}}{3 x}-\frac {a^{3} \ln \left (a x -1\right )}{6}+\frac {a^{3} \ln \left (a x +1\right )}{6}+\frac {2 a^{3} \dilog \left (a x \right )}{3}+\frac {2 a^{3} \dilog \left (a x +1\right )}{3}+\frac {2 a^{3} \ln \left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {a^{3} \ln \left (a x -1\right )^{2}}{6}-\frac {2 a^{3} \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{3}-\frac {a^{3} \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{3}-\frac {a^{3} \ln \left (a x +1\right )^{2}}{6}-\frac {a^{3} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{3}+\frac {a^{3} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 188, normalized size = 1.62 \[ -\frac {1}{6} \, {\left (4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 4 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a - a \log \left (a x + 1\right ) + a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} + 2}{x}\right )} a^{2} + \frac {1}{3} \, {\left (2 \, a^{2} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a \operatorname {artanh}\left (a x\right ) + \frac {{\left (3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{3 \, x^{3}} \]
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 \[ -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^4} \,d x \]
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 \[ - \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{4}}\right )\, dx - \int \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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