Integrand size = 14, antiderivative size = 222 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:
-1/2*arctan(((2*a^(1/2)*c^(1/2)-f)^(1/2)-2*c^(1/2)*x)/(2*a^(1/2)*c^(1/2)+f )^(1/2))/a^(1/2)/(2*a^(1/2)*c^(1/2)+f)^(1/2)+1/2*arctan(((2*a^(1/2)*c^(1/2 )-f)^(1/2)+2*c^(1/2)*x)/(2*a^(1/2)*c^(1/2)+f)^(1/2))/a^(1/2)/(2*a^(1/2)*c^ (1/2)+f)^(1/2)+1/2*arctanh((2*a^(1/2)*c^(1/2)-f)^(1/2)*x/(a^(1/2)+c^(1/2)* x^2))/a^(1/2)/(2*a^(1/2)*c^(1/2)-f)^(1/2)
Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {f-\sqrt {-4 a c+f^2}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {-4 a c+f^2}} \] Input:
Integrate[(a + f*x^2 + c*x^4)^(-1),x]
Output:
(Sqrt[2]*Sqrt[c]*(ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[f - Sqrt[-4*a*c + f^2]]] /Sqrt[f - Sqrt[-4*a*c + f^2]] - ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[f + Sqrt[- 4*a*c + f^2]]]/Sqrt[f + Sqrt[-4*a*c + f^2]]))/Sqrt[-4*a*c + f^2]
Time = 0.85 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.64, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1407, 27, 1142, 25, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{a+c x^4+f x^2} \, dx\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-\sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-\sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\frac {1}{2} \sqrt {c} \int -\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\sqrt {2 \sqrt {a} \sqrt {c}-f} \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {1}{2} \sqrt {c} \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {1}{2} \sqrt {c} \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}\) |
Input:
Int[(a + f*x^2 + c*x^4)^(-1),x]
Output:
((Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]*ArcTan[(Sqrt[c]*(-(Sqrt[2*Sqrt[a]*Sq rt[c] - f]/Sqrt[c]) + 2*x))/Sqrt[2*Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*S qrt[c] + f] - (Sqrt[c]*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt[ c]*x^2])/2)/(2*Sqrt[a]*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]) + ((Sqrt[c]*Sq rt[2*Sqrt[a]*Sqrt[c] - f]*ArcTan[(Sqrt[c]*(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqr t[c] + 2*x))/Sqrt[2*Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f] + ( Sqrt[c]*Log[Sqrt[a] + Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt[c]*x^2])/2)/(2* Sqrt[a]*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.17
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +f \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} f}\right )}{2}\) | \(38\) |
| default | \(4 c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{4 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{4 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )\) | \(117\) |
Input:
int(1/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*sum(1/(2*_R^3*c+_R*f)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*f+a))
Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (157) = 314\).
Time = 0.09 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.16 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} + f}{4 \, a^{2} c - a f^{2}}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (4 \, a c - f^{2} + {\left (4 \, a^{2} c f - a f^{3}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}}\right )} \sqrt {\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} + f}{4 \, a^{2} c - a f^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} + f}{4 \, a^{2} c - a f^{2}}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (4 \, a c - f^{2} + {\left (4 \, a^{2} c f - a f^{3}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}}\right )} \sqrt {\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} + f}{4 \, a^{2} c - a f^{2}}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} - f}{4 \, a^{2} c - a f^{2}}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (4 \, a c - f^{2} - {\left (4 \, a^{2} c f - a f^{3}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}}\right )} \sqrt {-\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} - f}{4 \, a^{2} c - a f^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} - f}{4 \, a^{2} c - a f^{2}}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (4 \, a c - f^{2} - {\left (4 \, a^{2} c f - a f^{3}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}}\right )} \sqrt {-\frac {{\left (4 \, a^{2} c - a f^{2}\right )} \sqrt {-\frac {1}{4 \, a^{3} c - a^{2} f^{2}}} - f}{4 \, a^{2} c - a f^{2}}}\right ) \] Input:
integrate(1/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
1/2*sqrt(1/2)*sqrt(((4*a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a^2*f^2)) + f)/(4 *a^2*c - a*f^2))*log(2*c*x + sqrt(1/2)*(4*a*c - f^2 + (4*a^2*c*f - a*f^3)* sqrt(-1/(4*a^3*c - a^2*f^2)))*sqrt(((4*a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a ^2*f^2)) + f)/(4*a^2*c - a*f^2))) - 1/2*sqrt(1/2)*sqrt(((4*a^2*c - a*f^2)* sqrt(-1/(4*a^3*c - a^2*f^2)) + f)/(4*a^2*c - a*f^2))*log(2*c*x - sqrt(1/2) *(4*a*c - f^2 + (4*a^2*c*f - a*f^3)*sqrt(-1/(4*a^3*c - a^2*f^2)))*sqrt(((4 *a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a^2*f^2)) + f)/(4*a^2*c - a*f^2))) + 1/ 2*sqrt(1/2)*sqrt(-((4*a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a^2*f^2)) - f)/(4* a^2*c - a*f^2))*log(2*c*x + sqrt(1/2)*(4*a*c - f^2 - (4*a^2*c*f - a*f^3)*s qrt(-1/(4*a^3*c - a^2*f^2)))*sqrt(-((4*a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a ^2*f^2)) - f)/(4*a^2*c - a*f^2))) - 1/2*sqrt(1/2)*sqrt(-((4*a^2*c - a*f^2) *sqrt(-1/(4*a^3*c - a^2*f^2)) - f)/(4*a^2*c - a*f^2))*log(2*c*x - sqrt(1/2 )*(4*a*c - f^2 - (4*a^2*c*f - a*f^3)*sqrt(-1/(4*a^3*c - a^2*f^2)))*sqrt(-( (4*a^2*c - a*f^2)*sqrt(-1/(4*a^3*c - a^2*f^2)) - f)/(4*a^2*c - a*f^2)))
Time = 0.57 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.39 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c^{2} - 128 a^{2} c f^{2} + 16 a f^{4}\right ) + t^{2} \left (- 16 a c f + 4 f^{3}\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a^{2} c f - 8 t^{3} a f^{3} + 4 t a c - 2 t f^{2}}{c} \right )} \right )\right )} \] Input:
integrate(1/(c*x**4+f*x**2+a),x)
Output:
RootSum(_t**4*(256*a**3*c**2 - 128*a**2*c*f**2 + 16*a*f**4) + _t**2*(-16*a *c*f + 4*f**3) + c, Lambda(_t, _t*log(x + (32*_t**3*a**2*c*f - 8*_t**3*a*f **3 + 4*_t*a*c - 2*_t*f**2)/c)))
\[ \int \frac {1}{a+f x^2+c x^4} \, dx=\int { \frac {1}{c x^{4} + f x^{2} + a} \,d x } \] Input:
integrate(1/(c*x^4+f*x^2+a),x, algorithm="maxima")
Output:
integrate(1/(c*x^4 + f*x^2 + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 1038 vs. \(2 (157) = 314\).
Time = 0.35 (sec) , antiderivative size = 1038, normalized size of antiderivative = 4.68 \[ \int \frac {1}{a+f x^2+c x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
1/4*(16*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^2 - 4*sqrt(2)*sqrt( c*f + sqrt(-4*a*c + f^2)*c)*a*c^3 - 32*a^2*c^3 + 8*sqrt(2)*sqrt(c*f + sqrt (-4*a*c + f^2)*c)*a*c^2*f - 8*a*c^3*f - 8*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f^2 + sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^2 + 16*a *c^2*f^2 - 2*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c*f^3 + 2*c^2*f^3 + sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*f^4 - 2*c*f^4 + 4*sqrt(2)*sqrt(-4 *a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f - sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f + 2*sqrt(2)*sqrt(-4*a*c + f^2 )*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c*f^2 - sqrt(2)*sqrt(-4*a*c + f^2)*sqrt (c*f + sqrt(-4*a*c + f^2)*c)*f^3 + 8*(4*a*c - f^2)*a*c^2 + 2*(4*a*c - f^2) *c^2*f - 2*(4*a*c - f^2)*c*f^2)*arctan(2*sqrt(1/2)*x/sqrt((f + sqrt(-4*a*c + f^2))/c))/((16*a^3*c^2 - 4*a^2*c^3 + 8*a^2*c^2*f - 8*a^2*c*f^2 + a*c^2* f^2 - 2*a*c*f^3 + a*f^4)*abs(c)) + 1/4*(16*sqrt(2)*sqrt(c*f - sqrt(-4*a*c + f^2)*c)*a^2*c^2 - 4*sqrt(2)*sqrt(c*f - sqrt(-4*a*c + f^2)*c)*a*c^3 + 32* a^2*c^3 + 8*sqrt(2)*sqrt(c*f - sqrt(-4*a*c + f^2)*c)*a*c^2*f + 8*a*c^3*f - 8*sqrt(2)*sqrt(c*f - sqrt(-4*a*c + f^2)*c)*a*c*f^2 + sqrt(2)*sqrt(c*f - s qrt(-4*a*c + f^2)*c)*c^2*f^2 - 16*a*c^2*f^2 - 2*sqrt(2)*sqrt(c*f - sqrt(-4 *a*c + f^2)*c)*c*f^3 - 2*c^2*f^3 + sqrt(2)*sqrt(c*f - sqrt(-4*a*c + f^2)*c )*f^4 + 2*c*f^4 - 4*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f - sqrt(-4*a*c + f^ 2)*c)*a*c*f + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f - sqrt(-4*a*c + f^2)*...
Time = 18.00 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.44 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=-\mathrm {atan}\left (\frac {f^4\,x\,1{}\mathrm {i}-f\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,c\,f^2\,x\,8{}\mathrm {i}}{64\,a^3\,c^2\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-f^3+4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}+4\,a\,f^4\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-f^3+4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}-32\,a^2\,c\,f^2\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-f^3+4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}}\right )\,\sqrt {\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-f^3+4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {f^4\,x\,1{}\mathrm {i}+f\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,c\,f^2\,x\,8{}\mathrm {i}}{4\,a\,f^4\,\sqrt {-\frac {f^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}+64\,a^3\,c^2\,\sqrt {-\frac {f^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}-32\,a^2\,c\,f^2\,\sqrt {-\frac {f^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}}\right )\,\sqrt {-\frac {f^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,c^2\,f^2-12\,a\,c\,f^4+f^6}-4\,a\,c\,f}{128\,a^3\,c^2-64\,a^2\,c\,f^2+8\,a\,f^4}}\,2{}\mathrm {i} \] Input:
int(1/(a + c*x^4 + f*x^2),x)
Output:
- atan((f^4*x*1i - f*x*(f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1 /2)*1i + a^2*c^2*x*16i - a*c*f^2*x*8i)/(64*a^3*c^2*(((f^6 - 64*a^3*c^3 + 4 8*a^2*c^2*f^2 - 12*a*c*f^4)^(1/2) - f^3 + 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2) + 4*a*f^4*(((f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12 *a*c*f^4)^(1/2) - f^3 + 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^( 1/2) - 32*a^2*c*f^2*(((f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1/ 2) - f^3 + 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2)))*(((f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1/2) - f^3 + 4*a*c*f)/(8*a*f ^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2)*2i - atan((f^4*x*1i + f*x*(f^6 - 6 4*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1/2)*1i + a^2*c^2*x*16i - a*c*f^ 2*x*8i)/(4*a*f^4*(-(f^3 + (f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4) ^(1/2) - 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2) + 64*a^3*c ^2*(-(f^3 + (f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1/2) - 4*a*c *f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2) - 32*a^2*c*f^2*(-(f^3 + (f^6 - 64*a^3*c^3 + 48*a^2*c^2*f^2 - 12*a*c*f^4)^(1/2) - 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c*f^2))^(1/2)))*(-(f^3 + (f^6 - 64*a^3*c^3 + 48*a^ 2*c^2*f^2 - 12*a*c*f^4)^(1/2) - 4*a*c*f)/(8*a*f^4 + 128*a^3*c^2 - 64*a^2*c *f^2))^(1/2)*2i
Time = 0.17 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.58 \[ \int \frac {1}{a+f x^2+c x^4} \, dx=\frac {2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a -2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f +\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a}{4 a \left (4 a c -f^{2}\right )} \] Input:
int(1/(c*x^4+f*x^2+a),x)
Output:
(2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f - 4*sqrt(c)*sqrt(2*sqrt(c)*sq rt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c) *sqrt(a) + f))*a - 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt (c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f + 4*sqrt(c) *sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c) *x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a - sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*l og( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f + sqrt(a)* sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f - 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt (c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a + 2*sqrt(c)*sqrt(2*sqrt(c)* sqrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a )/(4*a*(4*a*c - f**2))