Integrand size = 20, antiderivative size = 118 \[ \int \frac {1}{\sqrt {-a-b x^2-c x^4}} \, dx=\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {-a-b x^2-c x^4}} \] Output:
1/2*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^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 ))/a^(1/4)/c^(1/4)/(-c*x^4-b*x^2-a)^(1/2)
Result contains complex when optimal does not.
Time = 10.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {-a-b x^2-c x^4}} \, dx=-\frac {i \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b-\sqrt {b^2-4 a c}}} x\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {\frac {c}{b-\sqrt {b^2-4 a c}}} \sqrt {-a-x^2 \left (b+c x^2\right )}} \] Input:
Integrate[1/Sqrt[-a - b*x^2 - c*x^4],x]
Output:
((-I)*Sqrt[1 + (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 - x^2*(b + c*x^2)])
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1416}
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.
\(\displaystyle \int \frac {1}{\sqrt {-a-b x^2-c x^4}} \, dx\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {-a-b x^2-c x^4}}\) |
Input:
Int[1/Sqrt[-a - b*x^2 - c*x^4],x]
Output:
((Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^ 2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/ (2*a^(1/4)*c^(1/4)*Sqrt[-a - b*x^2 - c*x^4])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Time = 0.52 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20
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)*El lipticF(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 = 126, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {-a-b x^2-c x^4}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} \sqrt {-a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c})}{2 \, a c} \] Input:
integrate(1/(-c*x^4-b*x^2-a)^(1/2),x, algorithm="fricas")
Output:
1/2*sqrt(1/2)*(a*sqrt((b^2 - 4*a*c)/a^2) + b)*sqrt(-a)*sqrt((a*sqrt((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/(- a - b*x^2 - c*x^4)^(1/2),x)
Output:
int(1/(- a - b*x^2 - c*x^4)^(1/2), x)
\[ \int \frac {1}{\sqrt {-a-b x^2-c x^4}} \, dx=-\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) i \] 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)*i