Optimal. Leaf size=116 \[ -\frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{a^2}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{a^2}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{2 a^2}{3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3} \]
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Rubi [A] time = 0.423809, antiderivative size = 116, 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, 1612} \[ -\frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{a x+1}}\right )}-\frac{a^2}{2 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )}+\frac{a^2}{\left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^2}-\frac{2 a^2}{3 \left (\sqrt{\frac{1-a x}{a x+1}}+1\right )^3} \]
Antiderivative was successfully verified.
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Rule 6337
Rule 1612
Rubi steps
\begin{align*} \int \frac{e^{-\text{sech}^{-1}(a x)}}{x^3} \, dx &=\int \frac{1}{x^3 \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 (\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{(-1+x)^2 (1+x)^4} \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\left (\left (4 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{8 (-1+x)^2}-\frac{1}{2 (1+x)^4}+\frac{1}{2 (1+x)^3}-\frac{1}{8 (1+x)^2}\right ) \, dx,x,\sqrt{\frac{1-a x}{1+a x}}\right )\right )\\ &=-\frac{a^2}{2 \left (1-\sqrt{\frac{1-a x}{1+a x}}\right )}-\frac{2 a^2}{3 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^3}+\frac{a^2}{\left (1+\sqrt{\frac{1-a x}{1+a x}}\right )^2}-\frac{a^2}{2 \left (1+\sqrt{\frac{1-a x}{1+a x}}\right )}\\ \end{align*}
Mathematica [A] time = 0.0523257, size = 43, normalized size = 0.37 \[ -\frac{(a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2+1}{3 a x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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^{3}{\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 = 2.06645, size = 109, normalized size = 0.94 \begin{align*} -\frac{{\left (a^{3} x^{3} - a x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 1}{3 \, a 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*} a \int \frac{1}{a x^{3} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}} + x^{2}}\, 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^{3}{\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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