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*Sqrt[2]*Sqrt[k])

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Rubi [A]  time = 0.039761, 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, Rules used = {1107, 621, 204} \[ -\frac{\tan ^{-1}\left (\frac{e-2 k r^2}{\sqrt{2} \sqrt{k} \sqrt{-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt{2} \sqrt{k}} \]

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*Sqrt[2]*Sqrt[k])

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{r}{\sqrt{-\alpha ^2+2 e r^2-2 k r^4}} \, dr &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-\alpha ^2+2 e r-2 k r^2}} \, dr,r,r^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-8 k-r^2} \, dr,r,\frac{2 \left (e-2 k r^2\right )}{\sqrt{-\alpha ^2+2 e r^2-2 k r^4}}\right )\\ &=-\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}}\\ \end{align*}

Mathematica [A]  time = 0.0230724, size = 56, normalized size = 1. \[ -\frac{\tan ^{-1}\left (\frac{e-2 k r^2}{\sqrt{2} \sqrt{k} \sqrt{-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt{2} \sqrt{k}} \]

Antiderivative was successfully verified.

[In]

Integrate[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*Sqrt[2]*Sqrt[k])

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Maple [A]  time = 0.012, size = 47, normalized size = 0.8 \begin{align*}{\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}}}} \end{align*}

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))

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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(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69661, size = 381, normalized size = 6.8 \begin{align*} \left [-\frac{\sqrt{2} \sqrt{-k} \log \left (-8 \, k^{2} r^{4} + 8 \, e k r^{2} - 2 \, \alpha ^{2} k + 2 \, \sqrt{2} \sqrt{-2 \, k r^{4} + 2 \, e r^{2} - \alpha ^{2}}{\left (2 \, k r^{2} - e\right )} \sqrt{-k} - e^{2}\right )}{8 \, k}, -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, k r^{4} + 2 \, e r^{2} - \alpha ^{2}}{\left (2 \, k r^{2} - e\right )} \sqrt{k}}{2 \,{\left (2 \, k^{2} r^{4} - 2 \, e k r^{2} + \alpha ^{2} k\right )}}\right )}{4 \, \sqrt{k}}\right ] \end{align*}

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, algorithm="fricas")

[Out]

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

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

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)

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Giac [A]  time = 1.23936, size = 81, normalized size = 1.45 \begin{align*} -\frac{\sqrt{2} \log \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}} \end{align*}

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, algorithm="giac")

[Out]

-1/4*sqrt(2)*log(abs(sqrt(2)*(sqrt(2)*sqrt(-k)*r^2 - sqrt(-2*k*r^4 + 2*r^2*e - alpha^2))*sqrt(-k) + e))/sqrt(-
k)