Optimal. Leaf size=49 \[ -\frac{\text{sech}^{-1}(a x)^2}{x}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{x}-\frac{2}{x} \]
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Rubi [A] time = 0.0483221, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6285, 3296, 2638} \[ -\frac{\text{sech}^{-1}(a x)^2}{x}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{x}-\frac{2}{x} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a x)^2}{x^2} \, dx &=-\left (a \operatorname{Subst}\left (\int x^2 \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a x)^2}{x}+(2 a) \operatorname{Subst}\left (\int x \cosh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=\frac{2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{x}-\frac{\text{sech}^{-1}(a x)^2}{x}-(2 a) \operatorname{Subst}\left (\int \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{2}{x}+\frac{2 \sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{x}-\frac{\text{sech}^{-1}(a x)^2}{x}\\ \end{align*}
Mathematica [A] time = 0.0870271, size = 42, normalized size = 0.86 \[ -\frac{\text{sech}^{-1}(a x)^2-2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+2}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.213, size = 61, normalized size = 1.2 \begin{align*} a \left ( -{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{ax}}+2\,{\rm arcsech} \left (ax\right )\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}-2\,{\frac{1}{ax}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01563, size = 47, normalized size = 0.96 \begin{align*} 2 \, a \sqrt{\frac{1}{a^{2} x^{2}} - 1} \operatorname{arsech}\left (a x\right ) - \frac{\operatorname{arsech}\left (a x\right )^{2}}{x} - \frac{2}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70892, size = 208, normalized size = 4.24 \begin{align*} \frac{2 \, a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right ) - \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 2}{x} \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*} \int \frac{\operatorname{asech}^{2}{\left (a x \right )}}{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{\operatorname{arsech}\left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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