Optimal. Leaf size=35 \[ a \tanh ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )-\frac{e^{\text{sech}^{-1}(a x)}}{2 x} \]
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Rubi [B] time = 0.0412326, antiderivative size = 99, normalized size of antiderivative = 2.83, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6335, 30, 103, 12, 92, 208} \[ \frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{a x+1}}}+\frac{1}{2 a x^2}+\frac{1}{2} a \sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )-\frac{e^{\text{sech}^{-1}(a x)}}{x} \]
Warning: Unable to verify antiderivative.
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Rule 6335
Rule 30
Rule 103
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(a x)}}{x^2} \, dx &=-\frac{e^{\text{sech}^{-1}(a x)}}{x}-\frac{\int \frac{1}{x^3} \, dx}{a}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x^3 \sqrt{1-a x} \sqrt{1+a x}} \, dx}{a}\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{a^2}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 a}\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}-\frac{1}{2} \left (a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{2} \left (a^2 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )\\ &=\frac{1}{2 a x^2}-\frac{e^{\text{sech}^{-1}(a x)}}{x}+\frac{\sqrt{1-a x}}{2 a x^2 \sqrt{\frac{1}{1+a x}}}+\frac{1}{2} a \sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )\\ \end{align*}
Mathematica [B] time = 0.0598851, size = 93, normalized size = 2.66 \[ \frac{1}{2} \left (-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x^2}-\frac{1}{a x^2}-a \log (x)+a \log \left (a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.204, size = 90, normalized size = 2.6 \begin{align*} -{\frac{1}{2\,x}\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}}}}-{\frac{1}{2\,a{x}^{2}}} \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{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}}}{a} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81015, size = 278, normalized size = 7.94 \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*} \frac{\int \frac{1}{x^{3}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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