Optimal. Leaf size=48 \[ 2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )-\frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}} \]
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Rubi [A] time = 0.0308311, antiderivative size = 64, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6334, 97, 12, 41, 216} \[ -\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{a x+1}}}-\frac{1}{a x}-\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x) \]
Warning: Unable to verify antiderivative.
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Rule 6334
Rule 97
Rule 12
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x} \, dx &=-\frac{1}{a x}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{\sqrt{1-a x} \sqrt{1+a x}}{x^2} \, dx}{a}\\ &=-\frac{1}{a x}-\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{1+a x}}}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{a^2}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a}\\ &=-\frac{1}{a x}-\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{1+a x}}}-\left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=-\frac{1}{a x}-\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{1+a x}}}-\left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{a x}-\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{1+a x}}}-\sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \sin ^{-1}(a x)\\ \end{align*}
Mathematica [C] time = 0.0497251, size = 75, normalized size = 1.56 \[ \sqrt{\frac{1-a x}{a x+1}} \left (-\frac{1}{a x}-1\right )-\frac{1}{a x}-i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.189, size = 92, normalized size = 1.9 \begin{align*} -{{\it csgn} \left ( a \right ) \sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( \arctan \left ({{\it csgn} \left ( a \right ) ax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) xa+{\it csgn} \left ( a \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{ax}} \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*} \frac{-\frac{a^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{x}}{a} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02397, size = 165, normalized size = 3.44 \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*} \frac{\int \frac{1}{x^{2}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x}\, dx}{a} \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{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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