3.3.6 \(\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}} \, dr\) [206]

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

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {272, 65, 211} \begin {gather*} \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 {gather*}

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 65

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 211

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

Rule 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 46, normalized size = 1.00 \begin {gather*} \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 {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.57, size = 42, normalized size = 0.91 \begin {gather*} -\frac {\text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {\text {polar\_lift}\left [-\text {alpha}^2-\text {epsilon}^2\right ]}}{2 \sqrt {h} r}\right ]}{\sqrt {\text {polar\_lift}\left [-\text {alpha}^2-\text {epsilon}^2\right ]}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-ArcSinh[Sqrt[2] Sqrt[polar_lift[-alpha ^ 2 - epsilon ^ 2]] / (2 Sqrt[h] r)] / Sqrt[polar_lift[-alpha ^ 2 - ep
silon ^ 2]]

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Maple [A]
time = 0.04, size = 66, normalized size = 1.43

method	result	size
default	\(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}}\)	\(66\)

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,method=_RETURNVERBOSE)

[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 [A]
time = 0.35, size = 57, normalized size = 1.24 \begin {gather*} -\frac {\arcsin \left (\frac {\sqrt {2} \alpha ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} r} + \frac {\sqrt {2} \epsilon ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \end {gather*}

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]

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

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Fricas [A]
time = 0.30, size = 41, normalized size = 0.89 \begin {gather*} -\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 {gather*}

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 = 0.55, size = 42, normalized size = 0.91 \begin {gather*} - \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 {gather*}

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 = 0.00, size = 46, normalized size = 1.00 \begin {gather*} \frac {\left (5.0\times 10^{-13}\right )\cdot \left (2.0\times 10^{24}\right ) h\cdot 2.0\cdot 0.5 \arctan \left (\frac {\sqrt {-\left (1.0\times 10^{24}\right ) \alpha ^{2}+\left (2.0\times 10^{24}\right ) h r^{2}-1.0}}{\sqrt {\left (1.0\times 10^{24}\right ) \alpha ^{2}+1.0}}\right )}{h \sqrt {\left (1.0\times 10^{24}\right ) \alpha ^{2}+1.0}} \end {gather*}

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]

1.00000000000000e12*arctan(sqrt(2.00000000000000e24*h*r^2 - 1.00000000000000e24*alpha^2 - 1.00000000000000)/sq
rt(1.00000000000000e24*alpha^2 + 1.00000000000000))/sqrt(1.00000000000000e24*alpha^2 + 1.00000000000000)

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Mupad [B]
time = 0.66, size = 40, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,h\,r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \end {gather*}

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]

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

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