3.3.14 \(\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr\) [214]

3.3.14.1 Optimal result

Integrand size = 31, antiderivative size = 68 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=-\frac {\arctan \left (\frac {\alpha ^2+\epsilon ^2-h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}} \]

output

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

3.3.14.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {-k} r^2-\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]

input

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

output

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

3.3.14.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1434, 1154, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1154

\(\displaystyle -\int \frac {1}{-r^4-4 \left (\alpha ^2+\epsilon ^2\right )}d\left (-\frac {2 \left (\alpha ^2+\epsilon ^2-h r^2\right )}{\sqrt {-2 k r^4+2 h r^2-\alpha ^2-\epsilon ^2}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\arctan \left (\frac {\alpha ^2+\epsilon ^2-h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}}\)

input

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

output

-1/2*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])]/Sqrt[alpha^2 + epsilon^2]
 

3.3.14.3.1 Defintions of rubi rules used

rule 217

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-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]
 

rule 1434

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp 
[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
 

3.3.14.4 Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}+2 h \,r^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {-2 k \,r^{4}+2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r^{2}}\right )}{2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}}\)	\(78\)
elliptic	\(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}+2 h \,r^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {-2 k \,r^{4}+2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r^{2}}\right )}{2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}}\)	\(78\)
pseudoelliptic	\(-\frac {\ln \left (2\right )+\ln \left (\frac {h \,r^{2}+\sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {-2 k \,r^{4}+2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}-\alpha ^{2}-\epsilon ^{2}}{r^{2}}\right )}{2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}}\)	\(79\)

input

int(1/r/(-2*k*r^4+2*h*r^2-alpha^2-epsilon^2)^(1/2),r,method=_RETURNVERBOSE 
)
 

output

-1/2/(-alpha^2-epsilon^2)^(1/2)*ln((-2*alpha^2-2*epsilon^2+2*h*r^2+2*(-alp 
ha^2-epsilon^2)^(1/2)*(-2*k*r^4+2*h*r^2-alpha^2-epsilon^2)^(1/2))/r^2)
 

3.3.14.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.56 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=-\frac {\arctan \left (\frac {\sqrt {-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}} {\left (h r^{2} - \alpha ^{2} - \epsilon ^{2}\right )} \sqrt {\alpha ^{2} + \epsilon ^{2}}}{2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} k r^{4} + \alpha ^{4} + 2 \, \alpha ^{2} \epsilon ^{2} + \epsilon ^{4} - 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h r^{2}}\right )}{2 \, \sqrt {\alpha ^{2} + \epsilon ^{2}}} \]

input

integrate(1/r/(-2*k*r^4+2*h*r^2-alpha^2-epsilon^2)^(1/2),r, algorithm="fri 
cas")
 

output

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

3.3.14.6 Sympy [F]

\[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=\int \frac {1}{r \sqrt {- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r^{4}}}\, dr \]

input

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

output

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

3.3.14.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=\text {Exception raised: ValueError} \]

input

integrate(1/r/(-2*k*r^4+2*h*r^2-alpha^2-epsilon^2)^(1/2),r, algorithm="max 
ima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(2*epsilon^2*k+2*alpha^2*k>0)', s 
ee `assume
 

3.3.14.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=\frac {\arctan \left (-\frac {\sqrt {2} \sqrt {-k} r^{2} - \sqrt {-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}{\sqrt {\alpha ^{2} + \epsilon ^{2}}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]

input

integrate(1/r/(-2*k*r^4+2*h*r^2-alpha^2-epsilon^2)^(1/2),r, algorithm="gia 
c")
 

output

arctan(-(sqrt(2)*sqrt(-k)*r^2 - sqrt(-2*k*r^4 + 2*h*r^2 - alpha^2 - epsilo 
n^2))/sqrt(alpha^2 + epsilon^2))/sqrt(alpha^2 + epsilon^2)
 

3.3.14.9 Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06 \[ \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr=-\frac {\ln \left (h-\frac {\alpha ^2+\epsilon ^2}{r^2}+\frac {\sqrt {-\alpha ^2-\epsilon ^2}\,\sqrt {-\alpha ^2-\epsilon ^2-2\,k\,r^4+2\,h\,r^2}}{r^2}\right )}{2\,\sqrt {-\alpha ^2-\epsilon ^2}} \]

input

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

output

-log(h - (alpha^2 + epsilon^2)/r^2 + ((- alpha^2 - epsilon^2)^(1/2)*(2*h*r 
^2 - 2*k*r^4 - alpha^2 - epsilon^2)^(1/2))/r^2)/(2*(- alpha^2 - epsilon^2) 
^(1/2))