Optimal. Leaf size=130 \[ -\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.16, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6014, 6008, 266, 63, 208, 5950} \[ i a \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-i a \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 5950
Rule 6008
Rule 6014
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {\tanh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a}\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}+2 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.38, size = 121, normalized size = 0.93 \[ a \left (-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a x}+i \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-i \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+\log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{2}}, x\right ) \]
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 [A] time = 0.45, size = 188, normalized size = 1.45 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )}{x}-i a \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )-i a \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+i a \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )+i a \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+a \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )-a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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