Optimal. Leaf size=24 \[ -\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
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Rubi [A] time = 0.0416582, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6124, 844, 216, 266, 63, 208} \[ -\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6124
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{x} \, dx &=\int \frac{1-a x}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\left (a \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\right )+\int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\sin ^{-1}(a x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\sin ^{-1}(a x)-\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^2}\\ &=-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0146738, size = 28, normalized size = 1.17 \[ -\log \left (\sqrt{1-a^2 x^2}+1\right )-\sin ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.041, size = 93, normalized size = 3.9 \begin{align*} \sqrt{-{a}^{2}{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }-{a\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44148, size = 57, normalized size = 2.38 \begin{align*} -a{\left (\frac{\arcsin \left (a x\right )}{a} + \frac{\log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99666, size = 103, normalized size = 4.29 \begin{align*} 2 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21772, size = 70, normalized size = 2.92 \begin{align*} -\frac{a \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{a \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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