Optimal. Leaf size=90 \[ -\frac{1}{4} a^2 \text{sech}^{-1}(a x)^2-\frac{(1-a x) (a x+1)}{4 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^2}{2 x^2}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.0593064, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6285, 5372, 3310, 30} \[ -\frac{1}{4} a^2 \text{sech}^{-1}(a x)^2-\frac{(1-a x) (a x+1)}{4 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^2}{2 x^2}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5372
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a x)^2}{x^3} \, dx &=-\left (a^2 \operatorname{Subst}\left (\int x^2 \cosh (x) \sinh (x) \, dx,x,\text{sech}^{-1}(a x)\right )\right )\\ &=-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^2}{2 x^2}+a^2 \operatorname{Subst}\left (\int x \sinh ^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{(1-a x) (1+a x)}{4 x^2}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{2 x^2}-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^2}{2 x^2}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int x \, dx,x,\text{sech}^{-1}(a x)\right )\\ &=-\frac{(1-a x) (1+a x)}{4 x^2}+\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{2 x^2}-\frac{1}{4} a^2 \text{sech}^{-1}(a x)^2-\frac{(1-a x) (1+a x) \text{sech}^{-1}(a x)^2}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.041276, size = 54, normalized size = 0.6 \[ \frac{\left (a^2 x^2-2\right ) \text{sech}^{-1}(a x)^2+2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)-1}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.213, size = 77, normalized size = 0.9 \begin{align*}{a}^{2} \left ( -{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{2\,{a}^{2}{x}^{2}}}+{\frac{{\rm arcsech} \left (ax\right )}{2\,ax}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{ \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{4}}-{\frac{1}{4\,{a}^{2}{x}^{2}}} \right ) \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{\operatorname{arsech}\left (a x\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6307, size = 235, normalized size = 2.61 \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 ) +{\left (a^{2} x^{2} - 2\right )} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 1}{4 \, 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*} \int \frac{\operatorname{asech}^{2}{\left (a x \right )}}{x^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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