Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2+k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
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Rubi [A] time = 0.0564588, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2+k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(r*Sqrt[-alpha^2 - epsilon^2 - 2*k*r + 2*h*r^2]),r]
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Rubi in Sympy [A] time = 5.63441, size = 65, normalized size = 1.07 \[ - \frac{\operatorname{atan}{\left (- \frac{- 2 \alpha ^{2} - 2 \epsilon ^{2} - 2 k r}{2 \sqrt{\alpha ^{2} + \epsilon ^{2}} \sqrt{- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}} \right )}}{\sqrt{\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/r/(2*h*r**2-alpha**2-epsilon**2-2*k*r)**(1/2),r)
[Out]
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Mathematica [C] time = 0.144264, size = 72, normalized size = 1.18 \[ -\frac{i \log \left (\frac{2 \left (\sqrt{-\alpha ^2-\epsilon ^2+2 r (h r-k)}-\frac{i \left (\alpha ^2+\epsilon ^2+k r\right )}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{r}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(r*Sqrt[-alpha^2 - epsilon^2 - 2*k*r + 2*h*r^2]),r]
[Out]
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Maple [A] time = 0.004, size = 74, normalized size = 1.2 \[ -{1\ln \left ({\frac{1}{r} \left ( -2\,{\alpha }^{2}-2\,{\epsilon }^{2}-2\,kr+2\,\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}\sqrt{2\,h{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}-2\,kr} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/r/(2*h*r^2-alpha^2-epsilon^2-2*k*r)^(1/2),r)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)*r),r, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227985, size = 74, normalized size = 1.21 \[ -\frac{\arctan \left (\frac{\alpha ^{2} + \epsilon ^{2} + k r}{\sqrt{2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2} - 2 \, k r} \sqrt{\alpha ^{2} + \epsilon ^{2}}}\right )}{\sqrt{\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)*r),r, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{r \sqrt{- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}}\, dr \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/r/(2*h*r**2-alpha**2-epsilon**2-2*k*r)**(1/2),r)
[Out]
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GIAC/XCAS [A] time = 0.23366, size = 69, normalized size = 1.13 \[ \frac{2000000000000.0 \, \arctan \left (\frac{\left (6.5536 \times 10^{-08}\right ) \,{\left (-2.157918643757774 \times 10^{19} \, \sqrt{h} r + 1.52587890625 \times 10^{19} \, \sqrt{2.0 \, h r^{2} - \alpha ^{2} - 2.0 \, k r - 1 \times 10^{-24}}\right )}}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}}\right )}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)*r),r, algorithm="giac")
[Out]