\(\int \frac {1}{x^2 (a+f x^2+c x^4)} \, dx\) [788]

Optimal result

Integrand size = 18, antiderivative size = 264 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=-\frac {1}{a x}+\frac {\left (\sqrt {c}+\frac {f}{\sqrt {a}}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 a \sqrt {2 \sqrt {a} \sqrt {c}+f}}-\frac {\left (\sqrt {c}+\frac {f}{\sqrt {a}}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}+f}}\right )}{2 a \sqrt {2 \sqrt {a} \sqrt {c}+f}}+\frac {\left (\sqrt {c}-\frac {f}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}-f} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 a \sqrt {2 \sqrt {a} \sqrt {c}-f}} \] Output:

-1/a/x+1/2*(c^(1/2)+f/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*a^(1/2)*c^(1/2)+f)^(1/2)-1/2*(c^(1 
/2)+f/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*a^(1/2)*c^(1/2)+f)^(1/2)+1/2*(c^(1/2)-f/a^(1/2))*a 
rctanh((2*a^(1/2)*c^(1/2)-f)^(1/2)*x/(a^(1/2)+c^(1/2)*x^2))/a/(2*a^(1/2)*c 
^(1/2)-f)^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=-\frac {\frac {2}{x}+\frac {\sqrt {2} \sqrt {c} \left (f+\sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f-\sqrt {-4 a c+f^2}}}\right )}{\sqrt {-4 a c+f^2} \sqrt {f-\sqrt {-4 a c+f^2}}}+\frac {\sqrt {2} \sqrt {c} \left (-f+\sqrt {-4 a c+f^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {f+\sqrt {-4 a c+f^2}}}\right )}{\sqrt {-4 a c+f^2} \sqrt {f+\sqrt {-4 a c+f^2}}}}{2 a} \] Input:

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

Output:

-1/2*(2/x + (Sqrt[2]*Sqrt[c]*(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]]) + (Sqrt[2]*Sqrt[c]*(-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 + Sq 
rt[-4*a*c + f^2]]))/a
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.65, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1443, 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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1483

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-(1/(a*x)) - (((Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]*(Sqrt[a]*Sqrt[c] + f)* 
ArcTan[(Sqrt[c]*(-(Sqrt[2*Sqrt[a]*Sqrt[c] - f]/Sqrt[c]) + 2*x))/Sqrt[2*Sqr 
t[a]*Sqrt[c] + f]])/Sqrt[2*Sqrt[a]*Sqrt[c] + f] + (Sqrt[c]*(Sqrt[a]*Sqrt[c 
] - f)*Log[Sqrt[a] - Sqrt[2*Sqrt[a]*Sqrt[c] - f]*x + Sqrt[c]*x^2])/2)/(2*S 
qrt[a]*Sqrt[c]*Sqrt[2*Sqrt[a]*Sqrt[c] - f]) + ((Sqrt[c]*Sqrt[2*Sqrt[a]*Sqr 
t[c] - f]*(Sqrt[a]*Sqrt[c] + f)*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]*(Sqrt[a]*Sqrt[c] - f)*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
 

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]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {4 c \left (\frac {\left (-\sqrt {-4 a c +f^{2}}+f \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}\right )}{8 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}+f \right ) c}}-\frac {\left (-\sqrt {-4 a c +f^{2}}-f \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{8 \sqrt {-4 a c +f^{2}}\, \sqrt {\left (\sqrt {-4 a c +f^{2}}-f \right ) c}}\right )}{a}-\frac {1}{a x}\)	\(159\)
risch	\(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{5} c^{2}-8 c \,f^{2} a^{4}+f^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (12 a^{2} c^{2} f -7 a c \,f^{3}+f^{5}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2}-22 c \,f^{2} a^{4}+3 f^{4} a^{3}\right ) \textit {\_R}^{4}+\left (25 a^{2} c^{2} f -14 a c \,f^{3}+2 f^{5}\right ) \textit {\_R}^{2}+2 c^{3}\right ) x +\left (4 a^{4} c^{2}-5 a^{3} c \,f^{2}+a^{2} f^{4}\right ) \textit {\_R}^{3}\right )\right )}{2}\)	\(170\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1172 vs. \(2 (193) = 386\).

Time = 0.11 (sec) , antiderivative size = 1172, normalized size of antiderivative = 4.44 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(sqrt(1/2)*a*x*sqrt(-(3*a*c*f - f^3 + (4*a^4*c - a^3*f^2)*sqrt(-(a^2*c 
^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))/(4*a^4*c - a^3*f^2))*log(-2*(a 
*c^3 - c^2*f^2)*x + sqrt(1/2)*(4*a^2*c^2*f - 5*a*c*f^3 + f^5 - (8*a^5*c^2 
- 6*a^4*c*f^2 + a^3*f^4)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6* 
f^2)))*sqrt(-(3*a*c*f - f^3 + (4*a^4*c - a^3*f^2)*sqrt(-(a^2*c^2 - 2*a*c*f 
^2 + f^4)/(4*a^7*c - a^6*f^2)))/(4*a^4*c - a^3*f^2))) - sqrt(1/2)*a*x*sqrt 
(-(3*a*c*f - f^3 + (4*a^4*c - a^3*f^2)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/( 
4*a^7*c - a^6*f^2)))/(4*a^4*c - a^3*f^2))*log(-2*(a*c^3 - c^2*f^2)*x - sqr 
t(1/2)*(4*a^2*c^2*f - 5*a*c*f^3 + f^5 - (8*a^5*c^2 - 6*a^4*c*f^2 + a^3*f^4 
)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))*sqrt(-(3*a*c*f - 
 f^3 + (4*a^4*c - a^3*f^2)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^ 
6*f^2)))/(4*a^4*c - a^3*f^2))) + sqrt(1/2)*a*x*sqrt(-(3*a*c*f - f^3 - (4*a 
^4*c - a^3*f^2)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))/(4 
*a^4*c - a^3*f^2))*log(-2*(a*c^3 - c^2*f^2)*x + sqrt(1/2)*(4*a^2*c^2*f - 5 
*a*c*f^3 + f^5 + (8*a^5*c^2 - 6*a^4*c*f^2 + a^3*f^4)*sqrt(-(a^2*c^2 - 2*a* 
c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))*sqrt(-(3*a*c*f - f^3 - (4*a^4*c - a^3*f 
^2)*sqrt(-(a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))/(4*a^4*c - a^3 
*f^2))) - sqrt(1/2)*a*x*sqrt(-(3*a*c*f - f^3 - (4*a^4*c - a^3*f^2)*sqrt(-( 
a^2*c^2 - 2*a*c*f^2 + f^4)/(4*a^7*c - a^6*f^2)))/(4*a^4*c - a^3*f^2))*log( 
-2*(a*c^3 - c^2*f^2)*x - sqrt(1/2)*(4*a^2*c^2*f - 5*a*c*f^3 + f^5 + (8*...
 

Sympy [A] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} - 128 a^{4} c f^{2} + 16 a^{3} f^{4}\right ) + t^{2} \cdot \left (48 a^{2} c^{2} f - 28 a c f^{3} + 4 f^{5}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} + 48 t^{3} a^{4} c f^{2} - 8 t^{3} a^{3} f^{4} - 10 t a^{2} c^{2} f + 10 t a c f^{3} - 2 t f^{5}}{a c^{3} - c^{2} f^{2}} \right )} \right )\right )} - \frac {1}{a x} \] Input:

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

Output:

RootSum(_t**4*(256*a**5*c**2 - 128*a**4*c*f**2 + 16*a**3*f**4) + _t**2*(48 
*a**2*c**2*f - 28*a*c*f**3 + 4*f**5) + c**3, Lambda(_t, _t*log(x + (-64*_t 
**3*a**5*c**2 + 48*_t**3*a**4*c*f**2 - 8*_t**3*a**3*f**4 - 10*_t*a**2*c**2 
*f + 10*_t*a*c*f**3 - 2*_t*f**5)/(a*c**3 - c**2*f**2)))) - 1/(a*x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1857 vs. \(2 (193) = 386\).

Time = 0.66 (sec) , antiderivative size = 1857, normalized size of antiderivative = 7.03 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(8*a^3*c^3*f^2 - 2*a^2*c^2*f^4 - 4*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f 
 + sqrt(-4*a*c + f^2)*c)*a^3*c*f^2 + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + 
 sqrt(-4*a*c + f^2)*c)*a^2*c^2*f^2 - 2*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f 
 + sqrt(-4*a*c + f^2)*c)*a^2*c*f^3 + sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + 
 sqrt(-4*a*c + f^2)*c)*a^2*f^4 - 2*(4*a*c - f^2)*a^2*c^2*f^2 - (32*a^2*c^4 
 - 16*a*c^3*f^2 + 2*c^2*f^4 - 16*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqr 
t(-4*a*c + f^2)*c)*a^2*c^2 + 4*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt( 
-4*a*c + f^2)*c)*a*c^3 - 8*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a 
*c + f^2)*c)*a*c^2*f + 8*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c 
 + f^2)*c)*a*c*f^2 - sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f 
^2)*c)*c^2*f^2 + 2*sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2 
)*c)*c*f^3 - sqrt(2)*sqrt(-4*a*c + f^2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*f 
^4 - 8*(4*a*c - f^2)*a*c^3 + 2*(4*a*c - f^2)*c^2*f^2)*a^2 - 2*(16*sqrt(2)* 
sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^3*c^2*f - 4*sqrt(2)*sqrt(c*f + sqrt(-4* 
a*c + f^2)*c)*a^2*c^3*f - 32*a^3*c^3*f + 8*sqrt(2)*sqrt(c*f + sqrt(-4*a*c 
+ f^2)*c)*a^2*c^2*f^2 - 8*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a^2*c*f 
^3 + sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c^2*f^3 + 16*a^2*c^2*f^3 - 
 2*sqrt(2)*sqrt(c*f + sqrt(-4*a*c + f^2)*c)*a*c*f^4 + sqrt(2)*sqrt(c*f + s 
qrt(-4*a*c + f^2)*c)*a*f^5 - 2*a*c*f^5 + 8*(4*a*c - f^2)*a^2*c^2*f - 2*(4* 
a*c - f^2)*a*c*f^3)*abs(a))*arctan(2*sqrt(1/2)*x/sqrt((a*f + sqrt(-4*a^...
 

Mupad [B] (verification not implemented)

Time = 18.79 (sec) , antiderivative size = 2997, normalized size of antiderivative = 11.35 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((((4*a^4*c^2*f^3 - 16*a^5*c^3*f + x*(32*a^6*c^3*f - 8*a^5*c^2*f^3)* 
(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-( 
4*a*c - f^2)^3)^(1/2))/(8*(16*a^5*c^2 + a^3*f^4 - 8*a^4*c*f^2)))^(1/2))*(- 
(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4* 
a*c - f^2)^3)^(1/2))/(8*(16*a^5*c^2 + a^3*f^4 - 8*a^4*c*f^2)))^(1/2) + x*( 
4*a^4*c^4 - 2*a^3*c^3*f^2))*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2 
*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^5*c^2 + a^3*f^ 
4 - 8*a^4*c*f^2)))^(1/2)*1i + ((16*a^5*c^3*f - 4*a^4*c^2*f^3 + x*(32*a^6*c 
^3*f - 8*a^5*c^2*f^3)*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f 
 - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^5*c^2 + a^3*f^4 - 8* 
a^4*c*f^2)))^(1/2))*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^(1/2) + 12*a^2*c^2*f - 
 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^5*c^2 + a^3*f^4 - 8*a^ 
4*c*f^2)))^(1/2) + x*(4*a^4*c^4 - 2*a^3*c^3*f^2))*(-(f^5 + f^2*(-(4*a*c - 
f^2)^3)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/( 
8*(16*a^5*c^2 + a^3*f^4 - 8*a^4*c*f^2)))^(1/2)*1i)/(((16*a^5*c^3*f - 4*a^4 
*c^2*f^3 + x*(32*a^6*c^3*f - 8*a^5*c^2*f^3)*(-(f^5 + f^2*(-(4*a*c - f^2)^3 
)^(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16* 
a^5*c^2 + a^3*f^4 - 8*a^4*c*f^2)))^(1/2))*(-(f^5 + f^2*(-(4*a*c - f^2)^3)^ 
(1/2) + 12*a^2*c^2*f - 7*a*c*f^3 - a*c*(-(4*a*c - f^2)^3)^(1/2))/(8*(16*a^ 
5*c^2 + a^3*f^4 - 8*a^4*c*f^2)))^(1/2) + x*(4*a^4*c^4 - 2*a^3*c^3*f^2))...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.12 \[ \int \frac {1}{x^2 \left (a+f x^2+c x^4\right )} \, dx=\frac {4 \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 ) a c x -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^{2} x +2 \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 f x -4 \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 ) a c x +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^{2} x -2 \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 f x -2 \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 ) a c x +\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} x +2 \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 ) a c x -\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} x +\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 f x -\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 f x -16 a^{2} c +4 a \,f^{2}}{4 a^{2} x \left (4 a c -f^{2}\right )} \] Input:

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

Output:

(4*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*x - 2*sqrt(a)*sqrt(2*sqrt(c 
)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqr 
t(c)*sqrt(a) + f))*f**2*x + 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + f)*atan((sq 
rt(2*sqrt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*f* 
x - 4*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*x + 2*sqrt(a)*sqrt(2*sqr 
t(c)*sqrt(a) + f)*atan((sqrt(2*sqrt(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2* 
sqrt(c)*sqrt(a) + f))*f**2*x - 2*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*x - 2*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*x + 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*x 
+ 2*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*x - 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*x + sqrt(c)*s 
qrt(2*sqrt(c)*sqrt(a) - f)*log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) 
+ sqrt(c)*x**2)*a*f*x - sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - f)*log(sqrt(2*sqr 
t(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*f*x - 16*a**2*c + 4*a*f**2 
)/(4*a**2*x*(4*a*c - f**2))