Integrand size = 18, antiderivative size = 133 \[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt [4]{-a} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a-b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {-a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {-a+b x^2+c x^4}} \] Output:
1/2*(-a)^(1/4)*(1+c^(1/2)*x^2/(-a)^(1/2))*((-c*x^4-b*x^2+a)/a/(1+c^(1/2)*x ^2/(-a)^(1/2))^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/(-a)^(1/4)),1/2 *(2-b/(-a)^(1/2)/c^(1/2))^(1/2))/c^(1/4)/(c*x^4+b*x^2-a)^(1/2)
Result contains complex when optimal does not.
Time = 10.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=-\frac {i \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {-a+b x^2+c x^4}} \] Input:
Integrate[1/Sqrt[-a + b*x^2 + c*x^4],x]
Output:
((-I)*Sqrt[(b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt [1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c /(b + Sqrt[b^2 + 4*a*c])]*x], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a* c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[-a + b*x^2 + c*x^4])
Time = 0.53 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1417, 320}
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.
\(\displaystyle \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx\) |
\(\Big \downarrow \) 1417 |
\(\displaystyle \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \int \frac {1}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1}}dx}{\sqrt {-a+b x^2+c x^4}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {\sqrt {\sqrt {4 a c+b^2}+b} \left (\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {-a+b x^2+c x^4}}\) |
Input:
Int[1/Sqrt[-a + b*x^2 + c*x^4],x]
Output:
(Sqrt[b + Sqrt[b^2 + 4*a*c]]*(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))*Ellip ticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*Sqrt[(1 + (2*c*x^2)/ (b - Sqrt[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[-a + b*x^2 + c*x^4])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt[1 + 2*c*(x^2/(b + q ))]/Sqrt[a + b*x^2 + c*x^4]) Int[1/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Time = 0.52 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}\) | \(142\) |
elliptic | \(\frac {\sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}\) | \(142\) |
Input:
int(1/(c*x^4+b*x^2-a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*(4+2*(-b+(4*a*c+b^2)^(1/2))/a*x^2) ^(1/2)*(4-2*(b+(4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2-a)^(1/2)*Ellip ticF(1/2*x*(-2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^2)^ (1/2))/a/c)^(1/2))
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {-a} a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {-a} b\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c})}{2 \, a c} \] Input:
integrate(1/(c*x^4+b*x^2-a)^(1/2),x, algorithm="fricas")
Output:
-1/2*sqrt(1/2)*(sqrt(-a)*a*sqrt((b^2 + 4*a*c)/a^2) - sqrt(-a)*b)*sqrt((a*s qrt((b^2 + 4*a*c)/a^2) + b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*sqrt((a*sqrt( (b^2 + 4*a*c)/a^2) + b)/a)), 1/2*(a*b*sqrt((b^2 + 4*a*c)/a^2) - b^2 - 2*a* c)/(a*c))/(a*c)
\[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {- a + b x^{2} + c x^{4}}}\, dx \] Input:
integrate(1/(c*x**4+b*x**2-a)**(1/2),x)
Output:
Integral(1/sqrt(-a + b*x**2 + c*x**4), x)
\[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - a}} \,d x } \] Input:
integrate(1/(c*x^4+b*x^2-a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(c*x^4 + b*x^2 - a), x)
\[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - a}} \,d x } \] Input:
integrate(1/(c*x^4+b*x^2-a)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(c*x^4 + b*x^2 - a), x)
Timed out. \[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+b\,x^2-a}} \,d x \] Input:
int(1/(b*x^2 - a + c*x^4)^(1/2),x)
Output:
int(1/(b*x^2 - a + c*x^4)^(1/2), x)
\[ \int \frac {1}{\sqrt {-a+b x^2+c x^4}} \, dx=-\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{-c \,x^{4}-b \,x^{2}+a}d x \right ) \] Input:
int(1/(c*x^4+b*x^2-a)^(1/2),x)
Output:
- int(sqrt( - a + b*x**2 + c*x**4)/(a - b*x**2 - c*x**4),x)