Optimal. Leaf size=57 \[ \frac{2 a}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{4 a}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3} \]
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Rubi [A] time = 0.390062, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6337, 43} \[ \frac{2 a}{\left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^2}-\frac{4 a}{3 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )^3} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \text{sech}^{-1}(a x)}}{x^2} \, 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^2} \, dx\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \frac{x}{(-1+x)^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^4}+\frac{1}{(-1+x)^3}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{4 a}{3 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{2 a}{\left (1-\sqrt{\frac{1-a x}{1+a x}}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0662674, size = 52, normalized size = 0.91 \[ \frac{3 a^2 x^2+2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2-2}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 73, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}}{x}}-{\frac{1}{3\,{x}^{3}}} \right ) }+{\frac{2\,{a}^{2}{x}^{2}-2}{3\,a{x}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}-{\frac{1}{3\,{x}^{3}{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04345, size = 62, normalized size = 1.09 \begin{align*} \frac{1}{x} + \frac{2 \,{\left (a^{2} x^{3} - x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a^{2} x^{4}} - \frac{2}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05545, size = 130, normalized size = 2.28 \begin{align*} \frac{3 \, a^{2} x^{2} + 2 \,{\left (a^{3} x^{3} - a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 2}{3 \, a^{2} x^{3}} \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^{4}}\, dx + \int - \frac{a^{2}}{x^{2}}\, dx + \int \frac{2 a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{3}}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15194, size = 126, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (a^{2} + \frac{a}{x}\right )}^{\frac{3}{2}}{\left (a^{2} - \frac{a}{x}\right )} \sqrt{-a^{2} + \frac{a}{x}} a^{4} -{\left (a^{4} + \frac{a}{x^{3}}\right )} a^{6} +{\left (3 \,{\left (a^{2} + \frac{a}{x}\right )}^{2} a^{2} -{\left (a^{2} + \frac{a}{x}\right )}^{3}\right )} a^{4}}{3 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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