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

Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{h} r}{\sqrt{2 h r^2-\alpha ^2}}\right )}{\sqrt{2} \sqrt{h}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0114905, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{h} r}{\sqrt{2 h r^2-\alpha ^2}}\right )}{\sqrt{2} \sqrt{h}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-alpha^2 + 2*h*r^2],r]

[Out]

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0098647, size = 40, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{h} r}{\sqrt{2 h r^2-\alpha ^2}}\right )}{\sqrt{2} \sqrt{h}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 33, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}\ln \left ( \sqrt{h}r\sqrt{2}+\sqrt{2\,h{r}^{2}-{\alpha }^{2}} \right ){\frac{1}{\sqrt{h}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.60488, size = 235, normalized size = 5.88 \begin{align*} \left [\frac{\sqrt{2} \log \left (4 \, h r^{2} + 2 \, \sqrt{2} \sqrt{2 \, h r^{2} - \alpha ^{2}} \sqrt{h} r - \alpha ^{2}\right )}{4 \, \sqrt{h}}, -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{h}} \arctan \left (\frac{\sqrt{2} h r \sqrt{-\frac{1}{h}}}{\sqrt{2 \, h r^{2} - \alpha ^{2}}}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.19104, size = 68, normalized size = 1.7 \begin{align*} \begin{cases} \frac{\sqrt{2} \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{h} r}{\alpha } \right )}}{2 \sqrt{h}} & \text{for}\: \frac{2 \left |{h r^{2}}\right |}{\left |{\alpha ^{2}}\right |} > 1 \\- \frac{\sqrt{2} i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{h} r}{\alpha } \right )}}{2 \sqrt{h}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(2)*acosh(sqrt(2)*sqrt(h)*r/alpha)/(2*sqrt(h)), 2*Abs(h*r**2)/Abs(alpha**2) > 1), (-sqrt(2)*I*a
sin(sqrt(2)*sqrt(h)*r/alpha)/(2*sqrt(h)), True))

________________________________________________________________________________________

Giac [A]  time = 1.08898, size = 46, normalized size = 1.15 \begin{align*} -\frac{\sqrt{2} \log \left ({\left | -\sqrt{2} \sqrt{h} r + \sqrt{2 \, h r^{2} - \alpha ^{2}} \right |}\right )}{2 \, \sqrt{h}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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