Integrand size = 14, antiderivative size = 355 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (b^3-8 a b c-\frac {b^4-10 a b^2 c+56 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:
1/4*x*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/8*x*((-7*a*c+ b^2)*(-4*a*c+3*b^2)+3*b*c*(-8*a*c+b^2)*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^ 2+a)+3/16*c^(1/2)*(b^4-10*a*b^2*c+56*c^2*a^2+b*(-8*a*c+b^2)*(-4*a*c+b^2)^( 1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/( -4*a*c+b^2)^(5/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+3/16*c^(1/2)*(b^3-8*a*b*c-( 56*a^2*c^2-10*a*b^2*c+b^4)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b +(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a^2/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/ 2))^(1/2)
Time = 0.66 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 a x \left (b^2-2 a c+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 x \left (3 b^4-25 a b^2 c+28 a^2 c^2+3 b^3 c x^2-24 a b c^2 x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{16 a^2} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(-3),x]
Output:
((4*a*x*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ( 2*x*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2 + 3*b^3*c*x^2 - 24*a*b*c^2*x^2))/((b^ 2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(S qrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[ b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^ 2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqr t[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b ^2 - 4*a*c]]))/(16*a^2)
Time = 1.02 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1405, 25, 1492, 27, 1480, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 \left (b^4-9 a c b^2+c \left (b^2-8 a c\right ) x^2 b+28 a^2 c^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {b^4-9 a c b^2+c \left (b^2-8 a c\right ) x^2 b+28 a^2 c^2}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {3 \left (\frac {1}{2} c \left (\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+b \left (b^2-8 a c\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (b \left (b^2-8 a c\right )-\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx\right )}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\sqrt {c} \left (\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+b \left (b^2-8 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b \left (b^2-8 a c\right )-\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^(-3),x]
Output:
(x*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (( x*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b*c*(b^2 - 8*a*c)*x^2))/(2*a*(b^2 - 4 *a*c)*(a + b*x^2 + c*x^4)) + (3*((Sqrt[c]*(b*(b^2 - 8*a*c) + (b^4 - 10*a*b ^2*c + 56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b*( b^2 - 8*a*c) - (b^4 - 10*a*b^2*c + 56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[( Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^ 2 - 4*a*c]])))/(2*a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(\frac {-\frac {3 b \,c^{2} \left (8 a c -b^{2}\right ) x^{7}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (28 a^{2} c^{2}-49 a \,b^{2} c +6 b^{4}\right ) x^{5}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (4 a^{2} c^{2}+20 a \,b^{2} c -3 b^{4}\right ) x^{3}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (44 a^{2} c^{2}-37 a \,b^{2} c +5 b^{4}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {b c \left (8 a c -b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {28 a^{2} c^{2}-9 a \,b^{2} c +b^{4}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{16 a^{2}}\) | \(331\) |
| default | \(64 c^{3} \left (-\frac {\frac {\frac {3 \left (128 \sqrt {-4 a c +b^{2}}\, a^{3} b \,c^{3}-80 \sqrt {-4 a c +b^{2}}\, a^{2} b^{3} c^{2}+16 \sqrt {-4 a c +b^{2}}\, a \,b^{5} c -\sqrt {-4 a c +b^{2}}\, b^{7}+384 a^{4} c^{4}-352 a^{3} b^{2} c^{3}+120 a^{2} b^{4} c^{2}-18 a \,b^{6} c +b^{8}\right ) x^{3}}{64 a^{2} c^{3}}-\frac {\left (-704 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}+432 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}-84 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +5 \sqrt {-4 a c +b^{2}}\, b^{6}+320 a^{3} b \,c^{3}-240 a^{2} b^{3} c^{2}+60 a \,b^{5} c -5 b^{7}\right ) x}{64 a \,c^{3}}}{\left (x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right )^{2}}+\frac {3 \left (128 \sqrt {-4 a c +b^{2}}\, a^{3} b \,c^{3}-80 \sqrt {-4 a c +b^{2}}\, a^{2} b^{3} c^{2}+16 \sqrt {-4 a c +b^{2}}\, a \,b^{5} c -\sqrt {-4 a c +b^{2}}\, b^{7}+896 a^{4} c^{4}-608 a^{3} b^{2} c^{3}+152 a^{2} b^{4} c^{2}-18 a \,b^{6} c +b^{8}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{64 a^{2} c^{2} \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {\frac {3 \left (-128 \sqrt {-4 a c +b^{2}}\, a^{3} b \,c^{3}+80 \sqrt {-4 a c +b^{2}}\, a^{2} b^{3} c^{2}-16 \sqrt {-4 a c +b^{2}}\, a \,b^{5} c +\sqrt {-4 a c +b^{2}}\, b^{7}+384 a^{4} c^{4}-352 a^{3} b^{2} c^{3}+120 a^{2} b^{4} c^{2}-18 a \,b^{6} c +b^{8}\right ) x^{3}}{64 a^{2} c^{3}}-\frac {\left (704 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-432 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+84 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c -5 \sqrt {-4 a c +b^{2}}\, b^{6}+320 a^{3} b \,c^{3}-240 a^{2} b^{3} c^{2}+60 a \,b^{5} c -5 b^{7}\right ) x}{64 a \,c^{3}}}{\left (x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}}+\frac {3 \left (128 \sqrt {-4 a c +b^{2}}\, a^{3} b \,c^{3}-80 \sqrt {-4 a c +b^{2}}\, a^{2} b^{3} c^{2}+16 \sqrt {-4 a c +b^{2}}\, a \,b^{5} c -\sqrt {-4 a c +b^{2}}\, b^{7}-896 a^{4} c^{4}+608 a^{3} b^{2} c^{3}-152 a^{2} b^{4} c^{2}+18 a \,b^{6} c -b^{8}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{64 a^{2} c^{2} \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (4 a c -b^{2}\right )^{2}}\right )\) | \(920\) |
Input:
int(1/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
(-3/8*b*c^2*(8*a*c-b^2)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/8/a^2*c*(28*a ^2*c^2-49*a*b^2*c+6*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-1/8*b*(4*a^2*c^2+2 0*a*b^2*c-3*b^4)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/8*(44*a^2*c^2-37*a*b ^2*c+5*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x)/(c*x^4+b*x^2+a)^2+3/16/a^2*sum ((-b*c*(8*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2+(28*a^2*c^2-9*a*b^2*c+b ^4)/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+ _Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 4323 vs. \(2 (309) = 618\).
Time = 0.60 (sec) , antiderivative size = 4323, normalized size of antiderivative = 12.18 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(c*x**4+b*x**2+a)**3,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:
integrate(1/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*(3*(b^3*c^2 - 8*a*b*c^3)*x^7 + (6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*x ^5 + (3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*x^3 + (5*a*b^4 - 37*a^2*b^2*c + 44 *a^3*c^2)*x)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8 + a^4*b^4 - 8 *a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 1 6*a^5*b*c^2)*x^2) - 3/8*integrate(-(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^3*c - 8*a*b*c^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4* c^2)
Leaf count of result is larger than twice the leaf count of optimal. 2707 vs. \(2 (309) = 618\).
Time = 0.97 (sec) , antiderivative size = 2707, normalized size of antiderivative = 7.63 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
3/32*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 17*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b ^7*c - 2*b^8*c + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 26*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 34*a*b^6*c^2 + 2*b^7*c^2 - 368*sqrt(2)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 128*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^ 4*c^3 - 232*a^2*b^4*c^3 - 30*a*b^5*c^3 + 448*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^4 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 736*a^3*b^2*c^ 4 + 176*a^2*b^3*c^4 - 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^5 - 896*a^4*c^5 - 352*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*b^7 + 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*b^6*c - 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* a^2*b^3*c^2 - 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 *c^2 + 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b *c^3 + 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^ 2*c^3 + 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*...
Time = 21.91 (sec) , antiderivative size = 10979, normalized size of antiderivative = 30.93 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(a + b*x^2 + c*x^4)^3,x)
Output:
((x*(5*b^4 + 44*a^2*c^2 - 37*a*b^2*c))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c) ) + (x^5*(6*b^4*c + 28*a^2*c^3 - 49*a*b^2*c^2))/(8*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^3*(4*a^2*b*c^2 - 3*b^5 + 20*a*b^3*c))/(8*a^2*(b^4 + 16*a ^2*c^2 - 8*a*b^2*c)) + (3*c*x^7*(b^3*c - 8*a*b*c^2))/(8*a^2*(b^4 + 16*a^2* c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c* x^6) - atan(((((3*(7340032*a^9*c^9 - 256*a^2*b^14*c^2 + 7424*a^3*b^12*c^3 - 94208*a^4*b^10*c^4 + 675840*a^5*b^8*c^5 - 2949120*a^6*b^6*c^6 + 7798784* a^7*b^4*c^7 - 11534336*a^8*b^2*c^8))/(512*(a^4*b^12 + 4096*a^10*c^6 - 24*a ^5*b^10*c + 240*a^6*b^8*c^2 - 1280*a^7*b^6*c^3 + 3840*a^8*b^4*c^4 - 6144*a ^9*b^2*c^5)) - (x*(-(9*(b^19 + b^4*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^9 *b*c^9 + 769*a^2*b^15*c^2 - 8620*a^3*b^13*c^3 + 63440*a^4*b^11*c^4 - 31686 4*a^5*b^9*c^5 + 1069824*a^6*b^7*c^6 - 2343936*a^7*b^5*c^7 + 3010560*a^8*b^ 3*c^8 + 49*a^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 41*a*b^17*c - 11*a*b^2*c*(- (4*a*c - b^2)^15)^(1/2)))/(512*(a^5*b^20 + 1048576*a^15*c^10 - 40*a^6*b^18 *c + 720*a^7*b^16*c^2 - 7680*a^8*b^14*c^3 + 53760*a^9*b^12*c^4 - 258048*a^ 10*b^10*c^5 + 860160*a^11*b^8*c^6 - 1966080*a^12*b^6*c^7 + 2949120*a^13*b^ 4*c^8 - 2621440*a^14*b^2*c^9)))^(1/2)*(262144*a^9*b*c^7 - 256*a^4*b^11*c^2 + 5120*a^5*b^9*c^3 - 40960*a^6*b^7*c^4 + 163840*a^7*b^5*c^5 - 327680*a^8* b^3*c^6))/(32*(a^4*b^8 + 256*a^8*c^4 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256 *a^7*b^2*c^3)))*(-(9*(b^19 + b^4*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^...
Time = 3.74 (sec) , antiderivative size = 6794, normalized size of antiderivative = 19.14 \[ \int \frac {1}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(c*x^4+b*x^2+a)^3,x)
Output:
(264*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b*c**2 - 66*sqrt(a)*sqrt (2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s qrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**3*c + 528*sqrt(a)*sqrt(2*sqrt(c)*sqrt( a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sq rt(a) + b))*a**3*b**2*c**2*x**2 + 528*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)* atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**3*x**4 + 6*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2* sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**5 - 132*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**4*c*x**2 + 132*sqrt (a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt (c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**3*c**2*x**4 + 528*sqrt(a)*sqrt (2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s qrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c**3*x**6 + 264*sqrt(a)*sqrt(2*sqrt( c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sq rt(c)*sqrt(a) + b))*a**2*b*c**4*x**8 + 12*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a ) + b))*a*b**6*x**2 - 54*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2* sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**5...