Integrand size = 44, antiderivative size = 53 \[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {b d+a e} x}{\sqrt {d} \sqrt {a-b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {b d+a e}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 11.91 (sec) , antiderivative size = 416, normalized size of antiderivative = 7.85 \[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\frac {i \sqrt {2+\frac {4 c x^2}{-b+\sqrt {b^2-4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} x\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) d}{-a e+\sqrt {a} \sqrt {-4 c d^2+a e^2}},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} x\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (-b+\sqrt {b^2-4 a c}\right ) d}{a e+\sqrt {a} \sqrt {-4 c d^2+a e^2}},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} x\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )\right )}{2 \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} d \sqrt {a-b x^2+c x^4}} \]
((I/2)*Sqrt[2 + (4*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])] - EllipticP i[((b - Sqrt[b^2 - 4*a*c])*d)/(-(a*e) + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), 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])] - EllipticPi[((-b + Sqrt[b^2 - 4*a*c])*d)/(a* e + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), 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 [c/(-b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a - b*x^2 + c*x^4])
Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2537, 218}
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.
| \(\displaystyle \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx\) |
| \(\Big \downarrow \) 2537 |
| \(\displaystyle a \int \frac {1}{\frac {a (b d+a e) x^2}{c x^4-b x^2+a}+a d}d\frac {x}{\sqrt {c x^4-b x^2+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\arctan \left (\frac {x \sqrt {a e+b d}}{\sqrt {d} \sqrt {a-b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {a e+b d}}\) |
3.11.5.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Co eff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Simp[A Subst[Int[1/(d - (b*d - a*e )*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
Time = 2.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(-\frac {\arctan \left (\frac {d \sqrt {c \,x^{4}-b \,x^{2}+a}}{x \sqrt {\left (a e +b d \right ) d}}\right )}{\sqrt {\left (a e +b d \right ) d}}\) | \(46\) |
| elliptic | \(-\frac {\arctan \left (\frac {d \sqrt {c \,x^{4}-b \,x^{2}+a}}{x \sqrt {\left (a e +b d \right ) d}}\right )}{\sqrt {\left (a e +b d \right ) d}}\) | \(46\) |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {d \sqrt {c \,x^{4}-b \,x^{2}+a}}{x \sqrt {\left (a e +b d \right ) d}}\right )}{\sqrt {\left (a e +b d \right ) d}}\) | \(46\) |
Time = 7.66 (sec) , antiderivative size = 304, normalized size of antiderivative = 5.74 \[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\left [-\frac {\sqrt {-b d^{2} - a d e} \log \left (-\frac {c^{2} d^{2} x^{8} - 2 \, {\left (4 \, b c d^{2} + 3 \, a c d e\right )} x^{6} + {\left (8 \, a b d e + a^{2} e^{2} + 2 \, {\left (4 \, b^{2} + a c\right )} d^{2}\right )} x^{4} + a^{2} d^{2} - 2 \, {\left (4 \, a b d^{2} + 3 \, a^{2} d e\right )} x^{2} + 4 \, {\left (c d x^{5} - {\left (2 \, b d + a e\right )} x^{3} + a d x\right )} \sqrt {c x^{4} - b x^{2} + a} \sqrt {-b d^{2} - a d e}}{c^{2} d^{2} x^{8} + 2 \, a c d e x^{6} + 2 \, a^{2} d e x^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{4} + a^{2} d^{2}}\right )}{4 \, {\left (b d^{2} + a d e\right )}}, \frac {\arctan \left (\frac {2 \, \sqrt {c x^{4} - b x^{2} + a} \sqrt {b d^{2} + a d e} x}{c d x^{4} - {\left (2 \, b d + a e\right )} x^{2} + a d}\right )}{2 \, \sqrt {b d^{2} + a d e}}\right ] \]
[-1/4*sqrt(-b*d^2 - a*d*e)*log(-(c^2*d^2*x^8 - 2*(4*b*c*d^2 + 3*a*c*d*e)*x ^6 + (8*a*b*d*e + a^2*e^2 + 2*(4*b^2 + a*c)*d^2)*x^4 + a^2*d^2 - 2*(4*a*b* d^2 + 3*a^2*d*e)*x^2 + 4*(c*d*x^5 - (2*b*d + a*e)*x^3 + a*d*x)*sqrt(c*x^4 - b*x^2 + a)*sqrt(-b*d^2 - a*d*e))/(c^2*d^2*x^8 + 2*a*c*d*e*x^6 + 2*a^2*d* e*x^2 + (2*a*c*d^2 + a^2*e^2)*x^4 + a^2*d^2))/(b*d^2 + a*d*e), 1/2*arctan( 2*sqrt(c*x^4 - b*x^2 + a)*sqrt(b*d^2 + a*d*e)*x/(c*d*x^4 - (2*b*d + a*e)*x ^2 + a*d))/sqrt(b*d^2 + a*d*e)]
\[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=- \int \left (- \frac {a}{a d \sqrt {a - b x^{2} + c x^{4}} + a e x^{2} \sqrt {a - b x^{2} + c x^{4}} + c d x^{4} \sqrt {a - b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c x^{4}}{a d \sqrt {a - b x^{2} + c x^{4}} + a e x^{2} \sqrt {a - b x^{2} + c x^{4}} + c d x^{4} \sqrt {a - b x^{2} + c x^{4}}}\, dx \]
-Integral(-a/(a*d*sqrt(a - b*x**2 + c*x**4) + a*e*x**2*sqrt(a - b*x**2 + c *x**4) + c*d*x**4*sqrt(a - b*x**2 + c*x**4)), x) - Integral(c*x**4/(a*d*sq rt(a - b*x**2 + c*x**4) + a*e*x**2*sqrt(a - b*x**2 + c*x**4) + c*d*x**4*sq rt(a - b*x**2 + c*x**4)), x)
\[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\int { -\frac {c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt {c x^{4} - b x^{2} + a}} \,d x } \]
\[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\int { -\frac {c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt {c x^{4} - b x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {a-c x^4}{\sqrt {a-b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx=\int \frac {a-c\,x^4}{\left (c\,d\,x^4+a\,e\,x^2+a\,d\right )\,\sqrt {c\,x^4-b\,x^2+a}} \,d x \]