Optimal. Leaf size=23 \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
[Out]
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Rubi [A] time = 0.00983596, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
Antiderivative was successfully verified.
[In] Int[r/Sqrt[-alpha^2 + 2*e*r^2],r]
[Out]
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Rubi in Sympy [A] time = 1.39683, size = 15, normalized size = 0.65 \[ \frac{\sqrt{- \alpha ^{2} + 2 e r^{2}}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(r/(2*e*r**2-alpha**2)**(1/2),r)
[Out]
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Mathematica [A] time = 0.00633438, size = 23, normalized size = 1. \[ \frac{\sqrt{2 e r^2-\alpha ^2}}{2 e} \]
Antiderivative was successfully verified.
[In] Integrate[r/Sqrt[-alpha^2 + 2*e*r^2],r]
[Out]
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Maple [A] time = 0.002, size = 20, normalized size = 0.9 \[{\frac{1}{2\,e}\sqrt{2\,e{r}^{2}-{\alpha }^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(r/(2*e*r^2-alpha^2)^(1/2),r)
[Out]
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Maxima [A] time = 1.316, size = 26, normalized size = 1.13 \[ \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2078, size = 26, normalized size = 1.13 \[ \frac{\sqrt{2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="fricas")
[Out]
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Sympy [A] time = 0.764982, size = 29, normalized size = 1.26 \[ \begin{cases} \frac{\sqrt{- \alpha ^{2} + 2 e r^{2}}}{2 e} & \text{for}\: e \neq 0 \\\frac{r^{2}}{2 \sqrt{- \alpha ^{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(r/(2*e*r**2-alpha**2)**(1/2),r)
[Out]
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GIAC/XCAS [A] time = 0.200474, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, \sqrt{2 \, r^{2} e - \alpha ^{2}} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(r/sqrt(2*e*r^2 - alpha^2),r, algorithm="giac")
[Out]