Optimal. Leaf size=86 \[ \frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}}-\frac{2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \]
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Rubi [A] time = 0.445547, antiderivative size = 86, 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, 1629, 260} \[ \frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}}-\frac{2}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\log (a x+1)-2 \log \left (1-\sqrt{\frac{1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1629
Rule 260
Rubi steps
\begin{align*} \int \frac{e^{2 \text{sech}^{-1}(a x)}}{x} \, dx &=\int \frac{\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 )^2}{x} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{x (1+x)}{(-1+x)^3 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^3}+\frac{1}{2 (-1+x)^2}-\frac{1}{2 (-1+x)}+\frac{x}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{2}{1-\sqrt{\frac{1-a x}{1+a x}}}-2 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )+2 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{2}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{2}{1-\sqrt{\frac{1-a x}{1+a x}}}-\log (1+a x)-2 \log \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0658266, size = 86, normalized size = 1. \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a^2 x^2}-\frac{1}{a^2 x^2}+\log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )-2 \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.187, size = 97, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) -{\frac{1}{{a}^{2}{x}^{2}}}-{\frac{1}{ax}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( -{a}^{2}{x}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +\sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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{2 \,{\left (\frac{1}{2} \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} a^{2} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{2 \, x^{2}}\right )}}{a^{2}} - \frac{1}{a^{2} x^{2}} - \int \frac{1}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14903, size = 306, normalized size = 3.56 \begin{align*} \frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2}{2 \, a^{2} x^{2}} \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{2}{x^{3}}\, dx + \int - \frac{a^{2}}{x}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{2}}\, dx}{a^{2}} \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{{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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