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

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

[Out]

ArcTan[Sqrt[-alpha^2 - epsilon^2 + 2*h*r^2]/Sqrt[alpha^2 + epsilon^2]]/Sqrt[alpha^2 + epsilon^2]

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Rubi [A]  time = 0.0335206, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {266, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sqrt[-alpha^2 - epsilon^2 + 2*h*r^2]/Sqrt[alpha^2 + epsilon^2]]/Sqrt[alpha^2 + epsilon^2]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0112394, size = 46, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt{\alpha ^2+\epsilon ^2}}\right )}{\sqrt{\alpha ^2+\epsilon ^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sqrt[-alpha^2 - epsilon^2 + 2*h*r^2]/Sqrt[alpha^2 + epsilon^2]]/Sqrt[alpha^2 + epsilon^2]

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Maple [A]  time = 0.009, size = 66, normalized size = 1.4 \begin{align*} -{\ln \left ({\frac{1}{r} \left ( -2\,{\alpha }^{2}-2\,{\epsilon }^{2}+2\,\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}\sqrt{2\,h{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64199, size = 132, normalized size = 2.87 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{\alpha ^{2} + \epsilon ^{2}}}{\sqrt{2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}\right )}{\sqrt{\alpha ^{2} + \epsilon ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.26772, size = 42, normalized size = 0.91 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{\operatorname{polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}}{2 \sqrt{h} r} \right )}}{\sqrt{\operatorname{polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-asinh(sqrt(2)*sqrt(polar_lift(-alpha**2 - epsilon**2))/(2*sqrt(h)*r))/sqrt(polar_lift(-alpha**2 - epsilon**2)
)

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Giac [A]  time = 1.08102, size = 51, normalized size = 1.11 \begin{align*} \frac{1000000000000.0 \, \arctan \left (\frac{1000000000000.0 \, \sqrt{1.9999999999999998 \, h r^{2} - 0.9999999999999999 \, \alpha ^{2} - 1 \times 10^{-24}}}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}}\right )}{\sqrt{1 \times 10^{24} \, \alpha ^{2} + 1.0}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000000000000.0*arctan(1000000000000.0*sqrt(1.9999999999999998*h*r^2 - 0.9999999999999999*alpha^2 - 1e-24)/sqr
t((1e+24)*alpha^2 + 1.0))/sqrt((1e+24)*alpha^2 + 1.0)