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

Optimal. Leaf size=37 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+k r}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]

[Out]

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

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Rubi [A]  time = 0.0176463, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {724, 204} \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+k r}{\alpha \sqrt{-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0183538, size = 39, normalized size = 1.05 \[ \frac{\tan ^{-1}\left (\frac{-\alpha ^2-k r}{\alpha \sqrt{2 r (h r-k)-\alpha ^2}}\right )}{\alpha } \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 1.4 \begin{align*} -{\ln \left ({\frac{1}{r} \left ( -2\,{\alpha }^{2}-2\,kr+2\,\sqrt{-{\alpha }^{2}}\sqrt{2\,h{r}^{2}-{\alpha }^{2}-2\,kr} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69996, size = 140, normalized size = 3.78 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{2 \, h r^{2} - \alpha ^{2} - 2 \, k r}{\left (\alpha ^{2} + k r\right )}}{2 \, \alpha h r^{2} - \alpha ^{3} - 2 \, \alpha k r}\right )}{\alpha } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(2*h*r^2 - alpha^2 - 2*k*r)*(alpha^2 + k*r)/(2*alpha*h*r^2 - alpha^3 - 2*alpha*k*r))/alpha

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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}}\, dr \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11944, size = 54, normalized size = 1.46 \begin{align*} \frac{2 \, \arctan \left (-\frac{\sqrt{2} \sqrt{h} r - \sqrt{2 \, h r^{2} - \alpha ^{2} - 2 \, k r}}{\alpha }\right )}{\alpha } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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