Optimal. Leaf size=72 \[ \frac{a}{\sqrt{\frac{1-a x}{a x+1}}+1}-\frac{a}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+a \left (-\tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )\right ) \]
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Rubi [A] time = 0.382769, antiderivative size = 72, 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, 77, 207} \[ \frac{a}{\sqrt{\frac{1-a x}{a x+1}}+1}-\frac{a}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}+a \left (-\tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 6337
Rule 77
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-\text{sech}^{-1}(a x)}}{x^2} \, dx &=\int \frac{1}{x^2 \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\\ &=(4 a) \operatorname{Subst}\left (\int \frac{x}{(-1+x) (1+x)^3} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname{Subst}\left (\int \left (\frac{1}{2 (1+x)^3}-\frac{1}{4 (1+x)^2}+\frac{1}{4 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a}{1+\sqrt{\frac{1-a x}{1+a x}}}+a \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\\ &=-\frac{a}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}+\frac{a}{1+\sqrt{\frac{1-a x}{1+a x}}}-a \tanh ^{-1}\left (\sqrt{\frac{1-a x}{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0573269, size = 92, normalized size = 1.28 \[ \frac{1}{2} \left (\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x^2}-\frac{1}{a x^2}+a \log (x)-a \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^{2}{\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 = 2.02791, size = 279, normalized size = 3.88 \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 x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 2}{4 \, a 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*} a \int \frac{1}{a x^{2} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + x}\, 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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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