Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
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Rubi [A] time = 0.0335206, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {266, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{r \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}} \, dr &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{r \sqrt{-\alpha ^2-\epsilon ^2+2 h r}} \, dr,r,r^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{-\alpha ^2-\epsilon ^2}{2 h}+\frac{r^2}{2 h}} \, dr,r,\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}\right )}{2 h}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}}\\ \end{align*}
Mathematica [A] time = 0.0112394, size = 46, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 66, normalized size = 1.4 \begin{align*} -{\ln \left ({\frac{1}{r} \left ( -2\,{\alpha }^{2}-2\,{\epsilon }^{2}+2\,\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}\sqrt{2\,h{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64199, size = 132, normalized size = 2.87 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{\alpha ^{2} + \epsilon ^{2}}}{\sqrt{2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}\right )}{\sqrt{\alpha ^{2} + \epsilon ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.26772, size = 42, normalized size = 0.91 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{\operatorname{polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}}{2 \sqrt{h} r} \right )}}{\sqrt{\operatorname{polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08102, size = 51, normalized size = 1.11 \begin{align*} \frac{1000000000000.0 \, \arctan \left (\frac{1000000000000.0 \, \sqrt{1.9999999999999998 \, h r^{2} - 0.9999999999999999 \, \alpha ^{2} - 1 \times 10^{-24}}}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}}\right )}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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