\(\int \frac {1}{x^2 (a-b x^2+c x^4)} \, dx\) [825]

Optimal result

Integrand size = 19, antiderivative size = 172 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=-\frac {1}{a x}+\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/a/x+1/2*c^(1/2)*(1+b/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*x/(b-( 
-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/2*c^(1/ 
2)*(1-b/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2 
))^(1/2))*2^(1/2)/a/(b+(-4*a*c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=-\frac {\frac {2}{x}+\frac {\sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b+\sqrt {b^2-4 a c}}}}{2 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1443, 1480, 221}

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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {1}{a x}\)

Input:

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

Output:

-(1/(a*x)) + ((Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTanh[(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]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqr 
t[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a
 

Defintions of rubi rules used

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1480

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]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {4 c \left (-\frac {\left (b -\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-\sqrt {-4 a c +b^{2}}-b \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a}-\frac {1}{a x}\)	\(159\)
risch	\(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{5} c^{2}-8 a^{4} b^{2} c +b^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (-12 a^{2} b \,c^{2}+7 a \,b^{3} c -b^{5}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2}-22 a^{4} b^{2} c +3 b^{4} a^{3}\right ) \textit {\_R}^{4}+\left (-25 a^{2} b \,c^{2}+14 a \,b^{3} c -2 b^{5}\right ) \textit {\_R}^{2}+2 c^{3}\right ) x +\left (4 a^{4} c^{2}-5 a^{3} b^{2} c +a^{2} b^{4}\right ) \textit {\_R}^{3}\right )\right )}{2}\)	\(172\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (137) = 274\).

Time = 0.12 (sec) , antiderivative size = 1108, normalized size of antiderivative = 6.44 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} - 128 a^{4} b^{2} c + 16 a^{3} b^{4}\right ) + t^{2} \left (- 48 a^{2} b c^{2} + 28 a b^{3} c - 4 b^{5}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} + 48 t^{3} a^{4} b^{2} c - 8 t^{3} a^{3} b^{4} + 10 t a^{2} b c^{2} - 10 t a b^{3} c + 2 t b^{5}}{a c^{3} - b^{2} c^{2}} \right )} \right )\right )} - \frac {1}{a x} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

-integrate((c*x^2 - b)/(c*x^4 - b*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. 1877 vs. \(2 (137) = 274\).

Time = 0.69 (sec) , antiderivative size = 1877, normalized size of antiderivative = 10.91 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(2*a^2*b^4*c^2 - 8*a^3*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 
 sqrt(b^2 - 4*a*c)*c)*a^2*b^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sq 
rt(b^2 - 4*a*c)*c)*a^3*b^2*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqr 
t(b^2 - 4*a*c)*c)*a^2*b^3*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b 
^2 - 4*a*c)*c)*a^2*b^2*c^2 - 2*(b^2 - 4*a*c)*a^2*b^2*c^2 + (2*b^4*c^2 - 16 
*a*b^2*c^3 + 32*a^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 
 4*a*c)*c)*b^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c) 
*c)*a*b^2*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c) 
*b^3*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^2 
*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^ 
2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 4 
*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 - 2*(b^2 
 - 4*a*c)*b^2*c^2 + 8*(b^2 - 4*a*c)*a*c^3)*a^2 + 2*(sqrt(2)*sqrt(-b*c - sq 
rt(b^2 - 4*a*c)*c)*a*b^5 - 8*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^2* 
b^3*c + 2*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*a*b^5*c + 1 
6*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 8*sqrt(2)*sqrt(-b*c 
 - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c 
)*c)*a*b^3*c^2 - 16*a^2*b^3*c^2 - 4*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)* 
c)*a^2*b*c^3 + 32*a^3*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c + 8*(b^2 - 4*a*c)*a^ 
2*b*c^2)*abs(a))*arctan(2*sqrt(1/2)*x/sqrt(-(a*b + sqrt(a^2*b^2 - 4*a^3...
 

Mupad [B] (verification not implemented)

Time = 18.54 (sec) , antiderivative size = 2979, normalized size of antiderivative = 17.32 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.19 \[ \int \frac {1}{x^2 \left (a-b x^2+c x^4\right )} \, dx=\frac {4 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c x -2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2} x -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a b x -4 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c x +2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2} x +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a b x -2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c x +\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{2} x +2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c x -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{2} x -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a b x +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a b x -16 a^{2} c +4 a \,b^{2}}{4 a^{2} x \left (4 a c -b^{2}\right )} \] Input:

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

Output:

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