Optimal. Leaf size=21 \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]
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Rubi [A] time = 0.0161468, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {8} \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rubi steps
\begin{align*} \int \frac{1}{r \sqrt{-a^2+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2+2 H r^2}}\\ \end{align*}
Mathematica [A] time = 0.0000386, size = 21, normalized size = 1. \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931835, size = 26, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2}} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73989, size = 55, normalized size = 2.62 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2}} x}{2 \, H r^{3} - a^{2} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.052351, size = 15, normalized size = 0.71 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - a^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04571, size = 26, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2}} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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