3.211 \(\int \frac{r}{\sqrt{-\alpha ^2+2 e r^2-2 k r^4}} \, dr\)

Optimal. Leaf size=56 \[ -\frac{\tan ^{-1}\left (\frac{e-2 k r^2}{\sqrt{2} \sqrt{k} \sqrt{-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt{2} \sqrt{k}} \]

[Out]

-ArcTan[(e - 2*k*r^2)/(Sqrt[2]*Sqrt[k]*Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4])]/(2*S
qrt[2]*Sqrt[k])

_______________________________________________________________________________________

Rubi [A]  time = 0.0779085, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tan ^{-1}\left (\frac{e-2 k r^2}{\sqrt{2} \sqrt{k} \sqrt{-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt{2} \sqrt{k}} \]

Antiderivative was successfully verified.

[In]  Int[r/Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4],r]

[Out]

-ArcTan[(e - 2*k*r^2)/(Sqrt[2]*Sqrt[k]*Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4])]/(2*S
qrt[2]*Sqrt[k])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.3033, size = 56, normalized size = 1. \[ - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (2 e - 4 k r^{2}\right )}{4 \sqrt{k} \sqrt{- \alpha ^{2} + 2 e r^{2} - 2 k r^{4}}} \right )}}{4 \sqrt{k}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(r/(-2*k*r**4+2*e*r**2-alpha**2)**(1/2),r)

[Out]

-sqrt(2)*atan(sqrt(2)*(2*e - 4*k*r**2)/(4*sqrt(k)*sqrt(-alpha**2 + 2*e*r**2 - 2*
k*r**4)))/(4*sqrt(k))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0891028, size = 66, normalized size = 1.18 \[ \frac{i \log \left (2 \sqrt{-\alpha ^2+2 e r^2-2 k r^4}-\frac{i \sqrt{2} \left (2 k r^2-e\right )}{\sqrt{k}}\right )}{2 \sqrt{2} \sqrt{k}} \]

Antiderivative was successfully verified.

[In]  Integrate[r/Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4],r]

[Out]

((I/2)*Log[((-I)*Sqrt[2]*(-e + 2*k*r^2))/Sqrt[k] + 2*Sqrt[-alpha^2 + 2*e*r^2 - 2
*k*r^4]])/(Sqrt[2]*Sqrt[k])

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 47, normalized size = 0.8 \[{\frac{\sqrt{2}}{4}\arctan \left ({\sqrt{2}\sqrt{k} \left ({r}^{2}-{\frac{e}{2\,k}} \right ){\frac{1}{\sqrt{-2\,k{r}^{4}+2\,e{r}^{2}-{\alpha }^{2}}}}} \right ){\frac{1}{\sqrt{k}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r)

[Out]

1/4*2^(1/2)/k^(1/2)*arctan(2^(1/2)*k^(1/2)*(r^2-1/2*e/k)/(-2*k*r^4+2*e*r^2-alpha
^2)^(1/2))

_______________________________________________________________________________________

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(r/sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2),r, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.23133, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (-\sqrt{2}{\left (8 \, k^{2} r^{4} - 8 \, e k r^{2} + 2 \, \alpha ^{2} k + e^{2}\right )} \sqrt{-k} - 4 \, \sqrt{-2 \, k r^{4} + 2 \, e r^{2} - \alpha ^{2}}{\left (2 \, k^{2} r^{2} - e k\right )}\right )}{8 \, \sqrt{-k}}, \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (2 \, k r^{2} - e\right )}}{2 \, \sqrt{-2 \, k r^{4} + 2 \, e r^{2} - \alpha ^{2}} \sqrt{k}}\right )}{4 \, \sqrt{k}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2),r, algorithm="fricas")

[Out]

[1/8*sqrt(2)*log(-sqrt(2)*(8*k^2*r^4 - 8*e*k*r^2 + 2*alpha^2*k + e^2)*sqrt(-k) -
4*sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2)*(2*k^2*r^2 - e*k))/sqrt(-k), 1/4*sqrt(2)*a
rctan(1/2*sqrt(2)*(2*k*r^2 - e)/(sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2)*sqrt(k)))/sq
rt(k)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{r}{\sqrt{- \alpha ^{2} + 2 e r^{2} - 2 k r^{4}}}\, dr \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/(-2*k*r**4+2*e*r**2-alpha**2)**(1/2),r)

[Out]

Integral(r/sqrt(-alpha**2 + 2*e*r**2 - 2*k*r**4), r)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264474, size = 81, normalized size = 1.45 \[ -\frac{\sqrt{2}{\rm ln}\left ({\left | \sqrt{2}{\left (\sqrt{2} \sqrt{-k} r^{2} - \sqrt{-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}}\right )} \sqrt{-k} + e \right |}\right )}{4 \, \sqrt{-k}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(r/sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2),r, algorithm="giac")

[Out]

-1/4*sqrt(2)*ln(abs(sqrt(2)*(sqrt(2)*sqrt(-k)*r^2 - sqrt(-2*k*r^4 + 2*r^2*e - al
pha^2))*sqrt(-k) + e))/sqrt(-k)