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])

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Rubi [A]  time = 0.0255692, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{h} r}{\sqrt{2 h r^2-\alpha ^2}}\right )}{\sqrt{2} \sqrt{h}} \]

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])

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Rubi in Sympy [A]  time = 1.81254, size = 37, normalized size = 0.92 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{h} r}{\sqrt{- \alpha ^{2} + 2 h r^{2}}} \right )}}{2 \sqrt{h}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*h*r**2-alpha**2)**(1/2),r)

[Out]

sqrt(2)*atanh(sqrt(2)*sqrt(h)*r/sqrt(-alpha**2 + 2*h*r**2))/(2*sqrt(h))

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Mathematica [A]  time = 0.0147973, 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])

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Maple [A]  time = 0.002, size = 33, normalized size = 0.8 \[{\frac{\sqrt{2}}{2}\ln \left ( \sqrt{h}r\sqrt{2}+\sqrt{2\,h{r}^{2}-{\alpha }^{2}} \right ){\frac{1}{\sqrt{h}}}} \]

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*h*r^2 - alpha^2),r, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221142, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2} \log \left (4 \, h r^{2} + 2 \, \sqrt{2} \sqrt{2 \, h r^{2} - \alpha ^{2}} \sqrt{h} r - \alpha ^{2}\right )}{4 \, \sqrt{h}}, \frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{h}} \arctan \left (\frac{\sqrt{2} r}{\sqrt{2 \, h r^{2} - \alpha ^{2}} \sqrt{-\frac{1}{h}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*h*r^2 - alpha^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)*r/(sqrt(2*h*r^2 - alpha^2)*sqrt
(-1/h)))]

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Sympy [A]  time = 1.80004, size = 66, normalized size = 1.65 \[ \begin{cases} \frac{\sqrt{2} \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{h} r}{\alpha } \right )}}{2 \sqrt{h}} & \text{for}\: 2 \left |{\frac{h r^{2}}{\alpha ^{2}}}\right | > 1 \\- \frac{\sqrt{2} i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{h} r}{\alpha } \right )}}{2 \sqrt{h}} & \text{otherwise} \end{cases} \]

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/alph
a**2) > 1), (-sqrt(2)*I*asin(sqrt(2)*sqrt(h)*r/alpha)/(2*sqrt(h)), True))

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GIAC/XCAS [A]  time = 0.205761, size = 46, normalized size = 1.15 \[ -\frac{\sqrt{2}{\rm ln}\left ({\left | -\sqrt{2} \sqrt{h} r + \sqrt{2 \, h r^{2} - \alpha ^{2}} \right |}\right )}{2 \, \sqrt{h}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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