Optimal. Leaf size=174 \[ \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.39, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6014, 6020, 4182, 2531, 2282, 6589, 5994, 5950} \[ 2 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 5950
Rule 5994
Rule 6014
Rule 6020
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x} \, dx &=-\left (a^2 \int \frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-(2 a) \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\operatorname {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 \operatorname {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=4 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 \text {Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 203, normalized size = 1.17 \[ \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+2 i \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-2 i \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+2 \text {Li}_3\left (-e^{-\tanh ^{-1}(a x)}\right )-2 \text {Li}_3\left (e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+2 i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-2 i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\arctanh \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{x}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2}}{x} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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