Integrand size = 22, antiderivative size = 357 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\frac {\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}}{2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right )}-\frac {k \text {arctanh}\left (\frac {k-2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r}{\sqrt {2} \sqrt {\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}} \sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}}\right )}{2 \sqrt {2} \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right )^{3/2}} \] Output:
-1/4*k*arctanh(1/2*(-2*e*r+k)*2^(1/2)/e^(1/2)/(2*e*r^2-alpha^2-2*k*r)^(1/2 ))/e^(3/2)*2^(1/2)+1/2*(2*e*r^2-alpha^2-2*k*r)^(1/2)/e
Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.34 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\frac {\sqrt {-\alpha ^2+\frac {2 \text {g0} \text {g2L} \text {L2}^5 \left (\text {L2}^2+\text {L1} (5+\text {L2})\right )}{\text {L1} (\text {L1}+\text {L2})^6}-\frac {2 \left (-\text {g10} \text {g1L} \text {L2}^5+k (\text {L1}+\text {L2})^5 r\right )}{(\text {L1}+\text {L2})^5}} \left (\alpha ^2 \text {L1} (\text {L1}+\text {L2})^6-2 \text {g0} \text {g2L} \text {L2}^5 \left (\text {L2}^2+\text {L1} (5+\text {L2})\right )-\text {L1} (\text {L1}+\text {L2}) \left (2 \text {g10} \text {g1L} \text {L2}^5+k (\text {L1}+\text {L2})^5 r\right )\right )}{3 k^2 \text {L1} (\text {L1}+\text {L2})^6} \] Input:
Integrate[r/Sqrt[-alpha^2 - 2*k*r + 2*((5*g0*g2L*L2^5)/((L1 + L2)^6*r^2) + (g10*g1L*L2^5)/((L1 + L2)^5*r^2) + (g0*g2L*L2^6)/(L1*(L1 + L2)^5*r^2))*r^ 2],r]
Output:
(Sqrt[-alpha^2 + (2*g0*g2L*L2^5*(L2^2 + L1*(5 + L2)))/(L1*(L1 + L2)^6) - ( 2*(-(g10*g1L*L2^5) + k*(L1 + L2)^5*r))/(L1 + L2)^5]*(alpha^2*L1*(L1 + L2)^ 6 - 2*g0*g2L*L2^5*(L2^2 + L1*(5 + L2)) - L1*(L1 + L2)*(2*g10*g1L*L2^5 + k* (L1 + L2)^5*r)))/(3*k^2*L1*(L1 + L2)^6)
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.23, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1160, 1092, 219}
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.
\(\displaystyle \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2-2 k r}} \, dr\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {k \int \frac {1}{\sqrt {-\alpha ^2+2 e r^2-2 k r}}dr}{2 e}+\frac {\sqrt {-\alpha ^2+2 e r^2-2 k r}}{2 e}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {k \int \frac {1}{8 e-\frac {4 (k-2 e r)^2}{-\alpha ^2+2 e r^2-2 k r}}d\left (-\frac {2 (k-2 e r)}{\sqrt {-\alpha ^2+2 e r^2-2 k r}}\right )}{e}+\frac {\sqrt {-\alpha ^2+2 e r^2-2 k r}}{2 e}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {-\alpha ^2+2 e r^2-2 k r}}{2 e}-\frac {k \text {arctanh}\left (\frac {k-2 e r}{\sqrt {2} \sqrt {e} \sqrt {-\alpha ^2+2 e r^2-2 k r}}\right )}{2 \sqrt {2} e^{3/2}}\) |
Input:
Int[r/Sqrt[-alpha^2 - 2*k*r + 2*e*r^2],r]
Output:
Sqrt[-alpha^2 - 2*k*r + 2*e*r^2]/(2*e) - (k*ArcTanh[(k - 2*e*r)/(Sqrt[2]*S qrt[e]*Sqrt[-alpha^2 - 2*k*r + 2*e*r^2])])/(2*Sqrt[2]*e^(3/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.20
method | result | size |
default | \(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}}{2 e}+\frac {k \ln \left (\frac {\left (2 e r -k \right ) \sqrt {2}}{2 \sqrt {e}}+\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\right ) \sqrt {2}}{4 e^{\frac {3}{2}}}\) | \(70\) |
risch | \(-\frac {-2 e \,r^{2}+\alpha ^{2}+2 k r}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}}+\frac {k \ln \left (\frac {\left (2 e r -k \right ) \sqrt {2}}{2 \sqrt {e}}+\sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\right ) \sqrt {2}}{4 e^{\frac {3}{2}}}\) | \(84\) |
Input:
int(r/(2*e*r^2-alpha^2-2*k*r)^(1/2),r,method=_RETURNVERBOSE)
Output:
1/2*(2*e*r^2-alpha^2-2*k*r)^(1/2)/e+1/4*k/e^(3/2)*ln(1/2*(2*e*r-k)*2^(1/2) /e^(1/2)+(2*e*r^2-alpha^2-2*k*r)^(1/2))*2^(1/2)
Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.53 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\left [\frac {\sqrt {2} \sqrt {e} k \log \left (8 \, e^{2} r^{2} - 2 \, \alpha ^{2} e - 8 \, e k r + 2 \, \sqrt {2} \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} {\left (2 \, e r - k\right )} \sqrt {e} + k^{2}\right ) + 4 \, \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} e}{8 \, e^{2}}, -\frac {\sqrt {2} \sqrt {-e} k \arctan \left (\frac {\sqrt {2} \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} {\left (2 \, e r - k\right )} \sqrt {-e}}{2 \, {\left (2 \, e^{2} r^{2} - \alpha ^{2} e - 2 \, e k r\right )}}\right ) - 2 \, \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r} e}{4 \, e^{2}}\right ] \] Input:
integrate(r/(2*e*r^2-alpha^2-2*k*r)^(1/2),r, algorithm="fricas")
Output:
[1/8*(sqrt(2)*sqrt(e)*k*log(8*e^2*r^2 - 2*alpha^2*e - 8*e*k*r + 2*sqrt(2)* sqrt(2*e*r^2 - alpha^2 - 2*k*r)*(2*e*r - k)*sqrt(e) + k^2) + 4*sqrt(2*e*r^ 2 - alpha^2 - 2*k*r)*e)/e^2, -1/4*(sqrt(2)*sqrt(-e)*k*arctan(1/2*sqrt(2)*s qrt(2*e*r^2 - alpha^2 - 2*k*r)*(2*e*r - k)*sqrt(-e)/(2*e^2*r^2 - alpha^2*e - 2*e*k*r)) - 2*sqrt(2*e*r^2 - alpha^2 - 2*k*r)*e)/e^2]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (71) = 142\).
Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.48 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\begin {cases} \frac {k \left (\begin {cases} \frac {\sqrt {2} \log {\left (2 \sqrt {2} \sqrt {e} \sqrt {- \alpha ^{2} + 2 e r^{2} - 2 k r} + 4 e r - 2 k \right )}}{2 \sqrt {e}} & \text {for}\: \alpha ^{2} + \frac {k^{2}}{2 e} \neq 0 \\\frac {\sqrt {2} \left (r - \frac {k}{2 e}\right ) \log {\left (r - \frac {k}{2 e} \right )}}{2 \sqrt {e \left (r - \frac {k}{2 e}\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 e} + \frac {\sqrt {- \alpha ^{2} + 2 e r^{2} - 2 k r}}{2 e} & \text {for}\: e \neq 0 \\\frac {\alpha ^{2} \sqrt {- \alpha ^{2} - 2 k r} + \frac {\left (- \alpha ^{2} - 2 k r\right )^{\frac {3}{2}}}{3}}{2 k^{2}} & \text {for}\: k \neq 0 \\\frac {r^{2}}{2 \sqrt {- \alpha ^{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(r/(2*e*r**2-alpha**2-2*k*r)**(1/2),r)
Output:
Piecewise((k*Piecewise((sqrt(2)*log(2*sqrt(2)*sqrt(e)*sqrt(-alpha**2 + 2*e *r**2 - 2*k*r) + 4*e*r - 2*k)/(2*sqrt(e)), Ne(alpha**2 + k**2/(2*e), 0)), (sqrt(2)*(r - k/(2*e))*log(r - k/(2*e))/(2*sqrt(e*(r - k/(2*e))**2)), True ))/(2*e) + sqrt(-alpha**2 + 2*e*r**2 - 2*k*r)/(2*e), Ne(e, 0)), ((alpha**2 *sqrt(-alpha**2 - 2*k*r) + (-alpha**2 - 2*k*r)**(3/2)/3)/(2*k**2), Ne(k, 0 )), (r**2/(2*sqrt(-alpha**2)), True))
Exception generated. \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\text {Exception raised: ValueError} \] Input:
integrate(r/(2*e*r^2-alpha^2-2*k*r)^(1/2),r, algorithm="maxima")
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(k^2+2*alpha^2*e>0)', see `assume ?` for mor
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.21 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=-\frac {\sqrt {2} k \log \left ({\left | \sqrt {2} {\left (\sqrt {2} \sqrt {e} r - \sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r}\right )} \sqrt {e} - k \right |}\right )}{4 \, e^{\frac {3}{2}}} + \frac {\sqrt {2 \, e r^{2} - \alpha ^{2} - 2 \, k r}}{2 \, e} \] Input:
integrate(r/(2*e*r^2-alpha^2-2*k*r)^(1/2),r, algorithm="giac")
Output:
-1/4*sqrt(2)*k*log(abs(sqrt(2)*(sqrt(2)*sqrt(e)*r - sqrt(2*e*r^2 - alpha^2 - 2*k*r))*sqrt(e) - k))/e^(3/2) + 1/2*sqrt(2*e*r^2 - alpha^2 - 2*k*r)/e
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.19 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\frac {\sqrt {-\alpha ^2+2\,e\,r^2-2\,k\,r}}{2\,e}+\frac {\sqrt {2}\,k\,\ln \left (\sqrt {-\alpha ^2+2\,e\,r^2-2\,k\,r}-\frac {\sqrt {2}\,\left (k-2\,e\,r\right )}{2\,\sqrt {e}}\right )}{4\,e^{3/2}} \] Input:
int(r/(2*e*r^2 - 2*k*r - alpha^2)^(1/2),r)
Output:
(2*e*r^2 - 2*k*r - alpha^2)^(1/2)/(2*e) + (2^(1/2)*k*log((2*e*r^2 - 2*k*r - alpha^2)^(1/2) - (2^(1/2)*(k - 2*e*r))/(2*e^(1/2))))/(4*e^(3/2))
Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.22 \[ \int \frac {r}{\sqrt {-\alpha ^2-2 k r+2 \left (\frac {5 \text {g0} \text {g2L} \text {L2}^5}{(\text {L1}+\text {L2})^6 r^2}+\frac {\text {g10} \text {g1L} \text {L2}^5}{(\text {L1}+\text {L2})^5 r^2}+\frac {\text {g0} \text {g2L} \text {L2}^6}{\text {L1} (\text {L1}+\text {L2})^5 r^2}\right ) r^2}} \, dr=\frac {2 \sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\, e +\sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {2 e \,r^{2}-\alpha ^{2}-2 k r}\, \sqrt {2}+2 e r -k}{\sqrt {2 \alpha ^{2} e +k^{2}}}\right ) k}{4 e^{2}} \] Input:
int(r/(2*e*r^2-alpha^2-2*k*r)^(1/2),r)
Output:
(2*sqrt( - alpha**2 + 2*e*r**2 - 2*k*r)*e + sqrt(e)*sqrt(2)*log((sqrt(e)*s qrt( - alpha**2 + 2*e*r**2 - 2*k*r)*sqrt(2) + 2*e*r - k)/sqrt(2*alpha**2*e + k**2))*k)/(4*e**2)