3.213 \(\int \frac{1}{r \sqrt{-\alpha ^2+2 h r^2-2 k r^4}} \, dr\)

Optimal. Leaf size=44 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2-h r^2}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r^4}}\right )}{2 \alpha } \]

[Out]

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

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Rubi [A]  time = 0.0394127, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1114, 724, 204} \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2-h r^2}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r^4}}\right )}{2 \alpha } \]

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-alpha^2 + 2*h*r^2 - 2*k*r^4]),r]

[Out]

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

Rule 1114

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

Mathematica [A]  time = 0.0084803, size = 49, normalized size = 1.11 \[ \frac{\tan ^{-1}\left (\frac{2 h r^2-2 \alpha ^2}{2 \alpha \sqrt{-\alpha ^2+2 h r^2-2 k r^4}}\right )}{2 \alpha } \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-alpha^2 + 2*h*r^2 - 2*k*r^4]),r]

[Out]

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

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Maple [A]  time = 0.01, size = 56, normalized size = 1.3 \begin{align*} -{\frac{1}{2}\ln \left ({\frac{1}{{r}^{2}} \left ( -2\,{\alpha }^{2}+2\,h{r}^{2}+2\,\sqrt{-{\alpha }^{2}}\sqrt{-2\,k{r}^{4}+2\,h{r}^{2}-{\alpha }^{2}} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(-2*k*r^4+2*h*r^2-alpha^2)^(1/2),r)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61988, size = 155, normalized size = 3.52 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2}}{\left (h r^{2} - \alpha ^{2}\right )}}{2 \, \alpha k r^{4} - 2 \, \alpha h r^{2} + \alpha ^{3}}\right )}{2 \, \alpha } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*k*r^4+2*h*r^2-alpha^2)^(1/2),r, algorithm="fricas")

[Out]

-1/2*arctan(sqrt(-2*k*r^4 + 2*h*r^2 - alpha^2)*(h*r^2 - alpha^2)/(2*alpha*k*r^4 - 2*alpha*h*r^2 + alpha^3))/al
pha

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{r \sqrt{- \alpha ^{2} + 2 h r^{2} - 2 k r^{4}}}\, dr \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*k*r**4+2*h*r**2-alpha**2)**(1/2),r)

[Out]

Integral(1/(r*sqrt(-alpha**2 + 2*h*r**2 - 2*k*r**4)), r)

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Giac [A]  time = 1.22695, size = 42, normalized size = 0.95 \begin{align*} -\frac{\arcsin \left (-\frac{h - \frac{\alpha ^{2}}{r^{2}}}{\sqrt{-2 \, \alpha ^{2} k + h^{2}}}\right )}{2 \,{\left | \alpha \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*k*r^4+2*h*r^2-alpha^2)^(1/2),r, algorithm="giac")

[Out]

-1/2*arcsin(-(h - alpha^2/r^2)/sqrt(-2*alpha^2*k + h^2))/abs(alpha)