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

Optimal. Leaf size=68 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2-h r^2}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt{\alpha ^2+\epsilon ^2}} \]

[Out]

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

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Rubi [A]  time = 0.11392, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2-h r^2}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt{\alpha ^2+\epsilon ^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.44171, size = 65, normalized size = 0.96 \[ \frac{\operatorname{atan}{\left (\frac{- 2 \alpha ^{2} - 2 \epsilon ^{2} + 2 h r^{2}}{2 \sqrt{\alpha ^{2} + \epsilon ^{2}} \sqrt{- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r^{4}}} \right )}}{2 \sqrt{\alpha ^{2} + \epsilon ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/r/(-2*k*r**4+2*h*r**2-alpha**2-epsilon**2)**(1/2),r)

[Out]

atan((-2*alpha**2 - 2*epsilon**2 + 2*h*r**2)/(2*sqrt(alpha**2 + epsilon**2)*sqrt
(-alpha**2 - epsilon**2 + 2*h*r**2 - 2*k*r**4)))/(2*sqrt(alpha**2 + epsilon**2))

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Mathematica [C]  time = 0.30411, size = 80, normalized size = 1.18 \[ -\frac{i \log \left (\frac{2 \left (\sqrt{-\alpha ^2-\epsilon ^2+2 r^2 \left (h-k r^2\right )}-\frac{i \left (\alpha ^2+\epsilon ^2-h r^2\right )}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{r^2}\right )}{2 \sqrt{\alpha ^2+\epsilon ^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I/2)*Log[(2*(((-I)*(alpha^2 + epsilon^2 - h*r^2))/Sqrt[alpha^2 + epsilon^2] +
Sqrt[-alpha^2 - epsilon^2 + 2*r^2*(h - k*r^2)]))/r^2])/Sqrt[alpha^2 + epsilon^2
]

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Maple [A]  time = 0.011, size = 78, normalized size = 1.2 \[ -{\frac{1}{2}\ln \left ({\frac{1}{{r}^{2}} \left ( -2\,{\alpha }^{2}-2\,{\epsilon }^{2}+2\,h{r}^{2}+2\,\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}\sqrt{-2\,k{r}^{4}+2\,h{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227722, size = 85, normalized size = 1.25 \[ \frac{\arctan \left (\frac{h r^{2} - \alpha ^{2} - \epsilon ^{2}}{\sqrt{-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}} \sqrt{\alpha ^{2} + \epsilon ^{2}}}\right )}{2 \, \sqrt{\alpha ^{2} + \epsilon ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctan((h*r^2 - alpha^2 - epsilon^2)/(sqrt(-2*k*r^4 + 2*h*r^2 - alpha^2 - ep
silon^2)*sqrt(alpha^2 + epsilon^2)))/sqrt(alpha^2 + epsilon^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/r/(-2*k*r**4+2*h*r**2-alpha**2-epsilon**2)**(1/2),r)

[Out]

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

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GIAC/XCAS [A]  time = 0.260408, size = 42, normalized size = 0.62 \[ -\frac{\arcsin \left (-\frac{h - \frac{\alpha ^{2}}{r^{2}}}{\sqrt{-2 \, \alpha ^{2} k + h^{2}}}\right )}{2 \,{\left | \alpha \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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