Optimal. Leaf size=48 \[ -\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}-\frac {a x}{2}+\frac {1}{2} \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6014, 5912, 5916, 321, 206} \[ -\frac {1}{2} \text {PolyLog}(2,-a x)+\frac {1}{2} \text {PolyLog}(2,a x)-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac {a x}{2}+\frac {1}{2} \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 5912
Rule 5916
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x} \, dx &=-\left (a^2 \int x \tanh ^{-1}(a x) \, dx\right )+\int \frac {\tanh ^{-1}(a x)}{x} \, dx\\ &=-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}+\frac {1}{2} a^3 \int \frac {x^2}{1-a^2 x^2} \, dx\\ &=-\frac {a x}{2}-\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac {\text {Li}_2(-a x)}{2}+\frac {\text {Li}_2(a x)}{2}+\frac {1}{2} a \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a x}{2}+\frac {1}{2} \tanh ^{-1}(a x)-\frac {1}{2} a^2 x^2 \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 = 0.02, size = 60, normalized size = 1.25 \[ -\frac {1}{2} a^2 x^2 \tanh ^{-1}(a x)+\frac {1}{2} (\text {Li}_2(a x)-\text {Li}_2(-a x))-\frac {a x}{2}-\frac {1}{4} \log (1-a x)+\frac {1}{4} \log (a x+1) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{x}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 1.44 \[ -\frac {a^{2} x^{2} \arctanh \left (a x \right )}{2}+\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {a x}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 89, normalized size = 1.85 \[ -\frac {1}{4} \, a {\left (2 \, x + \frac {2 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (a^{2} x^{2} - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {\mathrm {atanh}\left (a\,x\right )\,\left (a^2\,x^2-1\right )}{x} \,d x \]
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 \[ - \int \left (- \frac {\operatorname {atanh}{\left (a x \right )}}{x}\right )\, dx - \int a^{2} x \operatorname {atanh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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