Optimal. Leaf size=152 \[ -a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-a^2 \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \left (-\tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.42, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6026, 6008, 266, 63, 208, 6020, 4182, 2531, 2282, 6589} \[ -a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+a^2 \text {PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-a^2 \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \left (-\tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2282
Rule 2531
Rule 4182
Rule 6008
Rule 6020
Rule 6026
Rule 6589
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a \int \frac {\tanh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-a^2 \operatorname {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \operatorname {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-a^2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-a^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-a^2 \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.27, size = 188, normalized size = 1.24 \[ \frac {1}{8} a^2 \left (8 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-8 \tanh ^{-1}(a x) \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+8 \text {Li}_3\left (-e^{-\tanh ^{-1}(a x)}\right )-8 \text {Li}_3\left (e^{-\tanh ^{-1}(a x)}\right )+4 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)+4 \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-4 \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )-4 \tanh ^{-1}(a x) \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+\tanh ^{-1}(a x)^2 \left (-\text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )-\tanh ^{-1}(a x)^2 \text {sech}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 231, normalized size = 1.52 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right ) \left (2 a x +\arctanh \left (a x \right )\right )}{2 x^{2}}-\frac {a^{2} \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-a^{2} \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{2} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{2} \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+a^{2} \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-a^{2} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a^{2} \arctanh \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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