\(\int \frac {1}{x^4 (a+f x^2+c x^4)} \, dx\) [789]

Optimal result

Integrand size = 18, antiderivative size = 305 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=-\frac {1}{3 a x^3}+\frac {f}{a^2 x}-\frac {\left (\sqrt {c} f-\frac {a c-f^2}{\sqrt {a}}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 a^2 \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\left (\sqrt {c} f-\frac {a c-f^2}{\sqrt {a}}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 a^2 \sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {\left (a c+\sqrt {a} \sqrt {c} f-f^2\right ) \text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 a^{5/2} \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:

-1/3/a/x^3+f/a^2/x-1/2*(c^(1/2)*f-(a*c-f^2)/a^(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^2/(2*a^(1/2)*c^ 
(1/2)+f)^(1/2)+1/2*(c^(1/2)*f-(a*c-f^2)/a^(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^2/(2*a^(1/2)*c^(1/2 
)+f)^(1/2)-1/2*(a*c+a^(1/2)*c^(1/2)*f-f^2)*arctanh((2*a^(1/2)*c^(1/2)-f)^( 
1/2)*x/(a^(1/2)+c^(1/2)*x^2))/a^(5/2)/(2*a^(1/2)*c^(1/2)-f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\frac {-\frac {2 a}{x^3}+\frac {6 f}{x}+\frac {3 \sqrt {2} \sqrt {c} \left (-2 a c+f \left (f+\sqrt {-4 a c+f^2}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {-4 a c+f^2} \sqrt {f-\sqrt {-4 a c+f^2}}}+\frac {3 \sqrt {2} \sqrt {c} \left (2 a c+f \left (-f+\sqrt {-4 a c+f^2}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {-4 a c+f^2} \sqrt {f+\sqrt {-4 a c+f^2}}}}{6 a^2} \] Input:

Integrate[1/(x^4*(a + f*x^2 + c*x^4)),x]
 

Output:

((-2*a)/x^3 + (6*f)/x + (3*Sqrt[2]*Sqrt[c]*(-2*a*c + f*(f + Sqrt[-4*a*c + 
f^2]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[f - Sqrt[-4*a*c + f^2]]])/(Sqrt[-4* 
a*c + f^2]*Sqrt[f - Sqrt[-4*a*c + f^2]]) + (3*Sqrt[2]*Sqrt[c]*(2*a*c + f*( 
-f + Sqrt[-4*a*c + f^2]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[f + Sqrt[-4*a*c 
+ f^2]]])/(Sqrt[-4*a*c + f^2]*Sqrt[f + Sqrt[-4*a*c + f^2]]))/(6*a^2)
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.58, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1443, 27, 1604, 25, 1483, 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.

\(\displaystyle \int \frac {1}{x^4 \left (a+c x^4+f x^2\right )} \, dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int -\frac {3 \left (c x^2+f\right )}{x^2 \left (c x^4+f x^2+a\right )}dx}{3 a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {c x^2+f}{x^2 \left (c x^4+f x^2+a\right )}dx}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 1604

\(\displaystyle -\frac {-\frac {\int -\frac {-f^2-c x^2 f+a c}{c x^4+f x^2+a}dx}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {-f^2-c x^2 f+a c}{c x^4+f x^2+a}dx}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 1483

\(\displaystyle -\frac {\frac {\frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )-\sqrt {c} \left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )+\sqrt {c} \left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )-\sqrt {c} \left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (a c-f^2\right )+\sqrt {c} \left (-f^2+\sqrt {a} \sqrt {c} f+a c\right ) x}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 1142

\(\displaystyle -\frac {\frac {\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int -\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {c} \left (x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c} \left (x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {\frac {\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx-\sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )^2-\frac {f+2 \sqrt {a} \sqrt {c}}{c}}d\left (2 x+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{x^2-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {1}{2} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \int \frac {2 \sqrt {c} x+\sqrt {2 \sqrt {a} \sqrt {c}-f}}{x^2+\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx+\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {\frac {\frac {\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (2 x-\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}+\frac {\frac {\sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f} \left (-\sqrt {a} \sqrt {c} f+a c-f^2\right ) \arctan \left (\frac {\sqrt {c} \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}}{\sqrt {c}}+2 x\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {1}{2} \sqrt {c} \left (\sqrt {a} \sqrt {c} f+a c-f^2\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}-f}+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-f}}}{a}-\frac {f}{a x}}{a}-\frac {1}{3 a x^3}\)

Input:

Int[1/(x^4*(a + f*x^2 + c*x^4)),x]
 

Output:

-1/3*1/(a*x^3) - (-(f/(a*x)) + (((Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(a*c 
 - Sqrt[a]*Sqrt[c]*f - f^2)*ArcTan[(Sqrt[c]*(-(Sqrt[2*Sqrt[a]*Sqrt[c] - f] 
/Sqrt[c]) + 2*x))/Sqrt[2*Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f 
] - (Sqrt[c]*(a*c + Sqrt[a]*Sqrt[c]*f - f^2)*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]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(a*c - Sqrt[a]*Sqrt[c]*f - 
 f^2)*ArcTan[(Sqrt[c]*(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqrt[c] + 2*x))/Sqrt[2* 
Sqrt[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f] + (Sqrt[c]*(a*c + Sqrt[ 
a]*Sqrt[c]*f - f^2)*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]))/a)/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1443

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim 
p[1/(a*d^2*(m + 1))   Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* 
x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 
- 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

rule 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {4 c \left (\frac {\left (f \sqrt {-4 a c +f^{2}}+2 a c -f^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{8 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\left (f \sqrt {-4 a c +f^{2}}-2 a c +f^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{8 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{a^{2}}-\frac {1}{3 a \,x^{3}}+\frac {f}{a^{2} x}\)	\(179\)
risch	\(\frac {\frac {f \,x^{2}}{a^{2}}-\frac {1}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 c^{2} a^{7}-8 a^{6} c \,f^{2}+a^{5} f^{4}\right ) \textit {\_Z}^{4}+\left (-20 a^{3} c^{3} f +25 a^{2} c^{2} f^{3}-9 a c \,f^{5}+f^{7}\right ) \textit {\_Z}^{2}+c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 c^{2} a^{7}-22 a^{6} c \,f^{2}+3 a^{5} f^{4}\right ) \textit {\_R}^{4}+\left (-43 a^{3} c^{3} f +51 a^{2} c^{2} f^{3}-18 a c \,f^{5}+2 f^{7}\right ) \textit {\_R}^{2}+2 c^{5}\right ) x +\left (-8 a^{5} c^{2} f +6 a^{4} c \,f^{3}-a^{3} f^{5}\right ) \textit {\_R}^{3}+a^{2} c^{4} \textit {\_R} \right )\right )}{2}\)	\(212\)

Input:

int(1/x^4/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

4/a^2*c*(1/8*(f*(-4*a*c+f^2)^(1/2)+2*a*c-f^2)/(-4*a*c+f^2)^(1/2)*2^(1/2)/( 
((-4*a*c+f^2)^(1/2)+f)*c)^(1/2)*arctan(c*x*2^(1/2)/(((-4*a*c+f^2)^(1/2)+f) 
*c)^(1/2))-1/8*(f*(-4*a*c+f^2)^(1/2)-2*a*c+f^2)/(-4*a*c+f^2)^(1/2)*2^(1/2) 
/(((-4*a*c+f^2)^(1/2)-f)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+f^2)^(1/2) 
-f)*c)^(1/2)))-1/3/a/x^3+f/a^2/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1654 vs. \(2 (230) = 460\).

Time = 0.11 (sec) , antiderivative size = 1654, normalized size of antiderivative = 5.42 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^4/(c*x^4+f*x^2+a),x, algorithm="fricas")
 

Output:

-1/6*(3*sqrt(1/2)*a^2*x^3*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 + (4*a^6*c - 
 a^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^ 
8)/(4*a^11*c - a^10*f^2)))/(4*a^6*c - a^5*f^2))*log(2*(a^2*c^5 - 3*a*c^4*f 
^2 + c^3*f^4)*x + sqrt(1/2)*(4*a^4*c^4 - 17*a^3*c^3*f^2 + 20*a^2*c^2*f^4 - 
 8*a*c*f^6 + f^8 + (12*a^7*c^2*f - 7*a^6*c*f^3 + a^5*f^5)*sqrt(-(a^4*c^4 - 
 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a^11*c - a^10*f^2))) 
*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 + (4*a^6*c - a^5*f^2)*sqrt(-(a^4*c^4 
- 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a^11*c - a^10*f^2)) 
)/(4*a^6*c - a^5*f^2))) - 3*sqrt(1/2)*a^2*x^3*sqrt((5*a^2*c^2*f - 5*a*c*f^ 
3 + f^5 + (4*a^6*c - a^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2* 
f^4 - 6*a*c*f^6 + f^8)/(4*a^11*c - a^10*f^2)))/(4*a^6*c - a^5*f^2))*log(2* 
(a^2*c^5 - 3*a*c^4*f^2 + c^3*f^4)*x - sqrt(1/2)*(4*a^4*c^4 - 17*a^3*c^3*f^ 
2 + 20*a^2*c^2*f^4 - 8*a*c*f^6 + f^8 + (12*a^7*c^2*f - 7*a^6*c*f^3 + a^5*f 
^5)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4* 
a^11*c - a^10*f^2)))*sqrt((5*a^2*c^2*f - 5*a*c*f^3 + f^5 + (4*a^6*c - a^5* 
f^2)*sqrt(-(a^4*c^4 - 6*a^3*c^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4 
*a^11*c - a^10*f^2)))/(4*a^6*c - a^5*f^2))) + 3*sqrt(1/2)*a^2*x^3*sqrt((5* 
a^2*c^2*f - 5*a*c*f^3 + f^5 - (4*a^6*c - a^5*f^2)*sqrt(-(a^4*c^4 - 6*a^3*c 
^3*f^2 + 11*a^2*c^2*f^4 - 6*a*c*f^6 + f^8)/(4*a^11*c - a^10*f^2)))/(4*a^6* 
c - a^5*f^2))*log(2*(a^2*c^5 - 3*a*c^4*f^2 + c^3*f^4)*x + sqrt(1/2)*(4*...
 

Sympy [A] (verification not implemented)

Time = 3.52 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{7} c^{2} - 128 a^{6} c f^{2} + 16 a^{5} f^{4}\right ) + t^{2} \left (- 80 a^{3} c^{3} f + 100 a^{2} c^{2} f^{3} - 36 a c f^{5} + 4 f^{7}\right ) + c^{5}, \left ( t \mapsto t \log {\left (x + \frac {- 96 t^{3} a^{7} c^{2} f + 56 t^{3} a^{6} c f^{3} - 8 t^{3} a^{5} f^{5} - 4 t a^{4} c^{4} + 32 t a^{3} c^{3} f^{2} - 40 t a^{2} c^{2} f^{4} + 16 t a c f^{6} - 2 t f^{8}}{a^{2} c^{5} - 3 a c^{4} f^{2} + c^{3} f^{4}} \right )} \right )\right )} + \frac {- a + 3 f x^{2}}{3 a^{2} x^{3}} \] Input:

integrate(1/x**4/(c*x**4+f*x**2+a),x)
 

Output:

RootSum(_t**4*(256*a**7*c**2 - 128*a**6*c*f**2 + 16*a**5*f**4) + _t**2*(-8 
0*a**3*c**3*f + 100*a**2*c**2*f**3 - 36*a*c*f**5 + 4*f**7) + c**5, Lambda( 
_t, _t*log(x + (-96*_t**3*a**7*c**2*f + 56*_t**3*a**6*c*f**3 - 8*_t**3*a** 
5*f**5 - 4*_t*a**4*c**4 + 32*_t*a**3*c**3*f**2 - 40*_t*a**2*c**2*f**4 + 16 
*_t*a*c*f**6 - 2*_t*f**8)/(a**2*c**5 - 3*a*c**4*f**2 + c**3*f**4)))) + (-a 
 + 3*f*x**2)/(3*a**2*x**3)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + f x^{2} + a\right )} x^{4}} \,d x } \] Input:

integrate(1/x^4/(c*x^4+f*x^2+a),x, algorithm="maxima")
 

Output:

integrate((c*f*x^2 - a*c + f^2)/(c*x^4 + f*x^2 + a), x)/a^2 + 1/3*(3*f*x^2 
 - a)/(a^2*x^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1664 vs. \(2 (230) = 460\).

Time = 0.82 (sec) , antiderivative size = 1664, normalized size of antiderivative = 5.46 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^4/(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^3*c^3 - 4*sqrt(2)*sqrt 
(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^4 - 32*a^3*c^4 + 8*sqrt(2)*sqrt(c*f + s 
qrt(-4*a*c + f^2)*c)*a^2*c^3*f - 24*a^2*c^4*f - 24*sqrt(2)*sqrt(c*f + sqrt 
(-4*a*c + f^2)*c)*a^2*c^2*f^2 + 5*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c) 
*a*c^3*f^2 + 48*a^2*c^3*f^2 - 10*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)* 
a*c^2*f^3 + 14*a*c^3*f^3 + 9*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c* 
f^4 - sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c^2*f^4 - 18*a*c^2*f^4 + 2* 
sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*c*f^5 - 2*c^2*f^5 - sqrt(2)*sqrt( 
c*f + sqrt(-4*a*c + f^2)*c)*f^6 + 2*c*f^6 + 12*sqrt(2)*sqrt(-4*a*c + f^2)* 
sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c^2*f - 3*sqrt(2)*sqrt(-4*a*c + f^2)* 
sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^3*f + 6*sqrt(2)*sqrt(-4*a*c + f^2)*sq 
rt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^2*f^2 - 7*sqrt(2)*sqrt(-4*a*c + f^2)*sq 
rt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f^3 + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c 
*f + sqrt(-4*a*c + f^2)*c)*c^2*f^3 - 2*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f 
 + sqrt(-4*a*c + f^2)*c)*c*f^4 + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqr 
t(-4*a*c + f^2)*c)*f^5 + 8*(4*a*c - f^2)*a^2*c^3 + 6*(4*a*c - f^2)*a*c^3*f 
 - 10*(4*a*c - f^2)*a*c^2*f^2 - 2*(4*a*c - f^2)*c^2*f^3 + 2*(4*a*c - f^2)* 
c*f^4)*arctan(2*sqrt(1/2)*x/sqrt((a^2*f + sqrt(-4*a^5*c + a^4*f^2))/(a^2*c 
)))/((16*a^5*c^2 - 4*a^4*c^3 + 8*a^4*c^2*f - 8*a^4*c*f^2 + a^3*c^2*f^2 - 2 
*a^3*c*f^3 + a^3*f^4)*abs(c)) - 1/4*(16*sqrt(2)*sqrt(c*f - sqrt(-4*a*c ...
 

Mupad [B] (verification not implemented)

Time = 18.52 (sec) , antiderivative size = 4162, normalized size of antiderivative = 13.65 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^4*(a + c*x^4 + f*x^2)),x)
 

Output:

- (1/(3*a) - (f*x^2)/a^2)/x^3 - atan(((x*(4*a^8*c^5 + 2*a^6*c^3*f^4 - 8*a^ 
7*c^4*f^2) + ((f^4*(-(4*a*c - f^2)^3)^(1/2) - f^7 + 20*a^3*c^3*f - 25*a^2* 
c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) + 9*a*c*f^5 - 3*a*c*f^2*(-(4*a* 
c - f^2)^3)^(1/2))/(8*(16*a^7*c^2 + a^5*f^4 - 8*a^6*c*f^2)))^(1/2)*(16*a^1 
0*c^4 - x*(32*a^11*c^3*f - 8*a^10*c^2*f^3)*((f^4*(-(4*a*c - f^2)^3)^(1/2) 
- f^7 + 20*a^3*c^3*f - 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) + 
 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^7*c^2 + a^5*f^4 
- 8*a^6*c*f^2)))^(1/2) + 4*a^8*c^2*f^4 - 20*a^9*c^3*f^2))*((f^4*(-(4*a*c - 
 f^2)^3)^(1/2) - f^7 + 20*a^3*c^3*f - 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - 
f^2)^3)^(1/2) + 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^7 
*c^2 + a^5*f^4 - 8*a^6*c*f^2)))^(1/2)*1i + (x*(4*a^8*c^5 + 2*a^6*c^3*f^4 - 
 8*a^7*c^4*f^2) - ((f^4*(-(4*a*c - f^2)^3)^(1/2) - f^7 + 20*a^3*c^3*f - 25 
*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1/2) + 9*a*c*f^5 - 3*a*c*f^2*(- 
(4*a*c - f^2)^3)^(1/2))/(8*(16*a^7*c^2 + a^5*f^4 - 8*a^6*c*f^2)))^(1/2)*(1 
6*a^10*c^4 + x*(32*a^11*c^3*f - 8*a^10*c^2*f^3)*((f^4*(-(4*a*c - f^2)^3)^( 
1/2) - f^7 + 20*a^3*c^3*f - 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a*c - f^2)^3)^(1 
/2) + 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^7*c^2 + a^5 
*f^4 - 8*a^6*c*f^2)))^(1/2) + 4*a^8*c^2*f^4 - 20*a^9*c^3*f^2))*((f^4*(-(4* 
a*c - f^2)^3)^(1/2) - f^7 + 20*a^3*c^3*f - 25*a^2*c^2*f^3 + a^2*c^2*(-(4*a 
*c - f^2)^3)^(1/2) + 9*a*c*f^5 - 3*a*c*f^2*(-(4*a*c - f^2)^3)^(1/2))/(8...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x^4 \left (a+f x^2+c x^4\right )} \, dx =\text {Too large to display} \] Input:

int(1/x^4/(c*x^4+f*x^2+a),x)
 

Output:

( - 18*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))*a*c*f*x**3 + 6*sqrt(a)*sqrt 
(2*sqrt(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/s 
qrt(2*sqrt(c)*sqrt(a) + f))*f**3*x**3 + 12*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**2*c*x**3 - 6*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*f**2*x* 
*3 + 18*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))*a*c*f*x**3 - 6*sqrt(a)*sqr 
t(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**3*x**3 - 12*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**2*c*x**3 + 6*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*f**2*x 
**3 + 9*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) 
- f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f*x**3 - 3*sqrt(a)*sqrt(2*sqrt(c)*sqr 
t(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f 
**3*x**3 - 9*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a 
) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f*x**3 + 3*sqrt(a)*sqrt(2*sqrt(c)*s 
qrt(a) - f)*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)...