\(\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr\) [208]

Optimal result

Integrand size = 29, antiderivative size = 61 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {738, 210} \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]

[In]

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

[Out]

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

Rule 210

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{4 \left (-\alpha ^2-\epsilon ^2\right )-r^2} \, dr,r,\frac {2 \left (-\alpha ^2-\epsilon ^2\right )-2 k r}{\sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )\right ) \\ & = -\frac {\arctan \left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {2 \arctan \left (\frac {\sqrt {2} \sqrt {h} r-\sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}-2 k r +2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}}\)	\(74\)

[In]

int(1/r/(2*h*r^2-alpha^2-epsilon^2-2*k*r)^(1/2),r,method=_RETURNVERBOSE)

[Out]

-1/(-alpha^2-epsilon^2)^(1/2)*ln((-2*alpha^2-2*epsilon^2-2*k*r+2*(-alpha^2-epsilon^2)^(1/2)*(2*h*r^2-alpha^2-e
psilon^2-2*k*r)^(1/2))/r)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.59 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {\arctan \left (-\frac {\sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2} - 2 \, k r} {\left (\alpha ^{2} + \epsilon ^{2} + k r\right )} \sqrt {\alpha ^{2} + \epsilon ^{2}}}{\alpha ^{4} + 2 \, \alpha ^{2} \epsilon ^{2} + \epsilon ^{4} - 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h r^{2} + 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} k r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]

[In]

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

[Out]

-arctan(-sqrt(2*h*r^2 - alpha^2 - epsilon^2 - 2*k*r)*(alpha^2 + epsilon^2 + k*r)*sqrt(alpha^2 + epsilon^2)/(al
pha^4 + 2*alpha^2*epsilon^2 + epsilon^4 - 2*(alpha^2 + epsilon^2)*h*r^2 + 2*(alpha^2 + epsilon^2)*k*r))/sqrt(a
lpha^2 + epsilon^2)

Sympy [F]

\[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=\int \frac {1}{r \sqrt {- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}}\, dr \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {\arcsin \left (\frac {k}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}}} + \frac {\alpha ^{2}}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}} r} + \frac {\epsilon ^{2}}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}} r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=\frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {h} r - \sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2} - 2 \, k r}}{\sqrt {\alpha ^{2} + \epsilon ^{2}}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr=-\frac {\ln \left (\frac {\sqrt {-\alpha ^2-\epsilon ^2}\,\sqrt {-\alpha ^2-\epsilon ^2+2\,h\,r^2-2\,k\,r}}{r}-\frac {\alpha ^2+\epsilon ^2}{r}-k\right )}{\sqrt {-\alpha ^2-\epsilon ^2}} \]

[In]

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

[Out]

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