3.208 \(\int \frac{1}{r \sqrt{-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr\)

Optimal. Leaf size=61 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2+k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

[Out]

-(ArcTan[(alpha^2 + epsilon^2 + k*r)/(Sqrt[alpha^2 + epsilon^2]*Sqrt[-alpha^2 -
epsilon^2 - 2*k*r + 2*h*r^2])]/Sqrt[alpha^2 + epsilon^2])

_______________________________________________________________________________________

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]

[Out]

-(ArcTan[(alpha^2 + epsilon^2 + k*r)/(Sqrt[alpha^2 + epsilon^2]*Sqrt[-alpha^2 -
epsilon^2 - 2*k*r + 2*h*r^2])]/Sqrt[alpha^2 + epsilon^2])

_______________________________________________________________________________________

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]

-atan(-(-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)))/sqrt(alpha**2 + epsilon**2)

_______________________________________________________________________________________

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]

((-I)*Log[(2*(((-I)*(alpha^2 + epsilon^2 + k*r))/Sqrt[alpha^2 + epsilon^2] + Sqr
t[-alpha^2 - epsilon^2 + 2*r*(-k + h*r)]))/r])/Sqrt[alpha^2 + epsilon^2]

_______________________________________________________________________________________

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]

-1/(-alpha^2-epsilon^2)^(1/2)*ln((-2*alpha^2-2*epsilon^2-2*k*r+2*(-alpha^2-epsil
on^2)^(1/2)*(2*h*r^2-alpha^2-epsilon^2-2*k*r)^(1/2))/r)

_______________________________________________________________________________________

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]

Exception raised: ValueError

_______________________________________________________________________________________

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]

-arctan((alpha^2 + epsilon^2 + k*r)/(sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)
*sqrt(alpha^2 + epsilon^2)))/sqrt(alpha^2 + epsilon^2)

_______________________________________________________________________________________

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]

Integral(1/(r*sqrt(-alpha**2 - epsilon**2 + 2*h*r**2 - 2*k*r)), r)

_______________________________________________________________________________________

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]

2000000000000.0*arctan((6.5536e-08)*(-(2.157918643757774e+19)*sqrt(h)*r + (1.525
87890625e+19)*sqrt(2.0*h*r^2 - alpha^2 - 2.0*k*r - 1e-24))/sqrt((1e+24)*alpha^2
+ 1.0))/sqrt((1e+24)*alpha^2 + 1.0)