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 = 0.0237135, 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, Rules used = {724, 204} \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2+k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{r \sqrt{-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{4 \left (-\alpha ^2-\epsilon ^2\right )-r^2} \, dr,r,\frac{2 \left (-\alpha ^2-\epsilon ^2\right )-2 k r}{\sqrt{-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2+k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}}\\ \end{align*}

Mathematica [A]  time = 0.0292344, size = 65, normalized size = 1.07 \[ \frac{\tan ^{-1}\left (\frac{-\alpha ^2-\epsilon ^2-k r}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 r (h r-k)}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-alpha^2 - epsilon^2 - 2*k*r + 2*h*r^2]),r]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 74, normalized size = 1.2 \begin{align*} -{\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}}}}} \end{align*}

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-epsilon^2)^(1/2)*(2*h*r^2-alpha^2-e
psilon^2-2*k*r)^(1/2))/r)

________________________________________________________________________________________

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.

[In]

integrate(1/r/(2*h*r^2-alpha^2-epsilon^2-2*k*r)^(1/2),r, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.72665, size = 325, normalized size = 5.33 \begin{align*} -\frac{\arctan \left (-\frac{\sqrt{2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2} - 2 \, k r}{\left (\alpha ^{2} + \epsilon ^{2} + k r\right )} \sqrt{\alpha ^{2} + \epsilon ^{2}}}{\alpha ^{4} + 2 \, \alpha ^{2} \epsilon ^{2} + \epsilon ^{4} - 2 \,{\left (\alpha ^{2} + \epsilon ^{2}\right )} h r^{2} + 2 \,{\left (\alpha ^{2} + \epsilon ^{2}\right )} k r}\right )}{\sqrt{\alpha ^{2} + \epsilon ^{2}}} \end{align*}

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, algorithm="fricas")

[Out]

-arctan(-sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)*(alpha^2 + epsilon^2 + k*r)*sqrt(alpha^2 + epsilon^2)/(al
pha^4 + 2*alpha^2*epsilon^2 + epsilon^4 - 2*(alpha^2 + epsilon^2)*h*r^2 + 2*(alpha^2 + epsilon^2)*k*r))/sqrt(a
lpha^2 + epsilon^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{r \sqrt{- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}}\, dr \end{align*}

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 [A]  time = 1.18963, size = 69, normalized size = 1.13 \begin{align*} \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}} \end{align*}

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, algorithm="giac")

[Out]

2000000000000.0*arctan((6.5536e-08)*(-(2.157918643757774e+19)*sqrt(h)*r + (1.52587890625e+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)