Optimal. Leaf size=46 \[ -\frac{2}{\sqrt{\frac{1-a x}{a x+1}}+1}-2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]
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Rubi [A] time = 0.401255, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6337, 801, 203} \[ -\frac{2}{\sqrt{\frac{1-a x}{a x+1}}+1}-2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6337
Rule 801
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{-\text{sech}^{-1}(a x)}}{x} \, dx &=\int \frac{1}{x \left (\frac{1}{a x}+\sqrt{\frac{1-a x}{1+a x}}+\frac{\sqrt{\frac{1-a x}{1+a x}}}{a x}\right )} \, dx\\ &=-\left (4 \operatorname{Subst}\left (\int \frac{x}{(1+x)^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left (4 \operatorname{Subst}\left (\int \left (-\frac{1}{2 (1+x)^2}+\frac{1}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{2}{1+\sqrt{\frac{1-a x}{1+a x}}}-2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2}{1+\sqrt{\frac{1-a x}{1+a x}}}-2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0358713, size = 74, normalized size = 1.61 \[ \sqrt{\frac{1-a x}{a x+1}} \left (\frac{1}{a x}+1\right )-\frac{1}{a x}+i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30327, size = 163, normalized size = 3.54 \begin{align*} \frac{a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - a x \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right ) - 1}{a x} \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*} a \int \frac{1}{a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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