Optimal. Leaf size=146 \[ -a x \tanh ^{-1}(a x)+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )+\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} \text {PolyLog}\left (3,-1+\frac {2}{1-a x}\right ) \]
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Rubi [A]
time = 0.24, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6161, 6033,
6199, 6095, 6205, 6745, 6037, 6127, 6021, 266} \begin {gather*} -\frac {1}{2} \log \left (1-a^2 x^2\right )-\frac {1}{2} 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)+2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2+\frac {1}{2} \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6021
Rule 6033
Rule 6037
Rule 6095
Rule 6127
Rule 6161
Rule 6199
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x} \, dx &=-\left (a^2 \int x \tanh ^{-1}(a x)^2 \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x} \, dx\\ &=-\frac {1}{2} a^2 x^2 \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+a^3 \int \frac {x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-a \int \tanh ^{-1}(a x) \, dx+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\\ &=-a x \tanh ^{-1}(a x)+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{2} a^2 x^2 \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 )+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+a^2 \int \frac {x}{1-a^2 x^2} \, dx\\ &=-a x \tanh ^{-1}(a x)+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \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 = 0.04, size = 145, normalized size = 0.99 \begin {gather*} -a x \tanh ^{-1}(a x)-\frac {1}{2} \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} \log \left (1-a^2 x^2\right )+\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {-1-a x}{-1+a x}\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {1+a x}{-1+a x}\right )-\frac {1}{2} \text {PolyLog}\left (3,\frac {-1-a x}{-1+a x}\right )+\frac {1}{2} \text {PolyLog}\left (3,\frac {1+a x}{-1+a x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 18.24, size = 663, normalized size = 4.54
method | result | size |
derivativedivides | \(-\frac {a^{2} x^{2} \arctanh \left (a x \right )^{2}}{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 )-2 \polylog \left (3, \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 )-2 \polylog \left (3, -\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 )+\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{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}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )}{2}+\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 )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}}{2}-\left (a x +1\right ) \arctanh \left (a x \right )+\frac {\arctanh \left (a x \right )^{2}}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {i \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 )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}\) | \(663\) |
default | \(-\frac {a^{2} x^{2} \arctanh \left (a x \right )^{2}}{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 )-2 \polylog \left (3, \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 )-2 \polylog \left (3, -\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 )+\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{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}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )}{2}+\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 )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}}{2}-\left (a x +1\right ) \arctanh \left (a x \right )+\frac {\arctanh \left (a x \right )^{2}}{2}+\ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {i \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 )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}\) | \(663\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x}\right )\, dx - \int a^{2} x \operatorname {atanh}^{2}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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