Optimal. Leaf size=53 \[ -\frac{\log (x)}{a^2}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2 \]
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Rubi [A] time = 0.0567211, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6285, 5418, 4184, 3475} \[ -\frac{\log (x)}{a^2}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5418
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \text{sech}^{-1}(a x)^2 \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \text{sech}^2(x) \tanh (x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^2}\\ &=\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2-\frac{\operatorname{Subst}\left (\int x \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2+\frac{\operatorname{Subst}\left (\int \tanh (x) \, dx,x,\text{sech}^{-1}(a x)\right )}{a^2}\\ &=-\frac{\sqrt{\frac{1-a x}{1+a x}} (1+a x) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2-\frac{\log (x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0595508, size = 53, normalized size = 1. \[ -\frac{\log (x)}{a^2}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.258, size = 101, normalized size = 1.9 \begin{align*} -{\frac{{\rm arcsech} \left (ax\right )}{{a}^{2}}}+{\frac{{x}^{2} \left ({\rm arcsech} \left (ax\right ) \right ) ^{2}}{2}}-{\frac{{\rm arcsech} \left (ax\right )x}{a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{1}{{a}^{2}}\ln \left ( 1+ \left ({\frac{1}{ax}}+\sqrt{{\frac{1}{ax}}-1}\sqrt{1+{\frac{1}{ax}}} \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04775, size = 54, normalized size = 1.02 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arsech}\left (a x\right )^{2} - \frac{x \sqrt{\frac{1}{a^{2} x^{2}} - 1} \operatorname{arsech}\left (a x\right )}{a} - \frac{\log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63633, size = 236, normalized size = 4.45 \begin{align*} \frac{a^{2} x^{2} \log \left (\frac{a x \sqrt{-\frac{a^{2} x^{2} - 1}{a^{2} x^{2}}} + 1}{a x}\right )^{2} - 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 ) - 2 \, \log \left (x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.11023, size = 42, normalized size = 0.79 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asech}^{2}{\left (a x \right )}}{2} - \frac{\sqrt{- a^{2} x^{2} + 1} \operatorname{asech}{\left (a x \right )}}{a^{2}} - \frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: a \neq 0 \\\infty x^{2} & \text{otherwise} \end{cases} \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 x \operatorname{arsech}\left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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