Optimal. Leaf size=37 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+k r}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]
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Rubi [A] time = 0.0176463, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {724, 204} \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+k r}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]
Antiderivative was successfully verified.
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Rule 724
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{r \sqrt{-\alpha ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-4 \alpha ^2-r^2} \, dr,r,\frac{-2 \alpha ^2-2 k r}{\sqrt{-\alpha ^2-2 k r+2 h r^2}}\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\alpha ^2+k r}{\alpha \sqrt{-\alpha ^2-2 k r+2 h r^2}}\right )}{\alpha }\\ \end{align*}
Mathematica [A] time = 0.0183538, size = 39, normalized size = 1.05 \[ \frac{\tan ^{-1}\left (\frac{-\alpha ^2-k r}{\alpha \sqrt{2 r (h r-k)-\alpha ^2}}\right )}{\alpha } \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 1.4 \begin{align*} -{\ln \left ({\frac{1}{r} \left ( -2\,{\alpha }^{2}-2\,kr+2\,\sqrt{-{\alpha }^{2}}\sqrt{2\,h{r}^{2}-{\alpha }^{2}-2\,kr} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{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 = 1.69996, size = 140, normalized size = 3.78 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{2 \, h r^{2} - \alpha ^{2} - 2 \, k r}{\left (\alpha ^{2} + k r\right )}}{2 \, \alpha h r^{2} - \alpha ^{3} - 2 \, \alpha k r}\right )}{\alpha } \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{1}{r \sqrt{- \alpha ^{2} + 2 h r^{2} - 2 k r}}\, dr \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11944, size = 54, normalized size = 1.46 \begin{align*} \frac{2 \, \arctan \left (-\frac{\sqrt{2} \sqrt{h} r - \sqrt{2 \, h r^{2} - \alpha ^{2} - 2 \, k r}}{\alpha }\right )}{\alpha } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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