\(\int \frac {1}{x^4 (a+b x^2+c x^4)} \, dx\) [775]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [B] (verification not implemented)
Sympy [A] (verification not implemented)
Maxima [F]
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 18, antiderivative size = 196 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{3 a x^3}+\frac {b}{a^2 x}+\frac {\sqrt {c} \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/3/a/x^3+b/a^2/x+1/2*c^(1/2)*(b+(-2*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/(b-(-4*a*c+b^2 
)^(1/2))^(1/2)+1/2*c^(1/2)*(b-(-2*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/(b+(-4*a*c+b^2)^(1 
/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {2 a}{x^3}+\frac {6 b}{x}+\frac {3 \sqrt {2} \sqrt {c} \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {2} \sqrt {c} \left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{6 a^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1443, 27, 1604, 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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {-\frac {\frac {\sqrt {c} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {b}{a x}}{a}-\frac {1}{3 a x^3}\)

Input:

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

Output:

-1/3*1/(a*x^3) - (-(b/(a*x)) - ((Sqrt[c]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a 
*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqr 
t[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b - (b^2 - 2*a*c)/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]]))/a)/a
 

Defintions of rubi rules used

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 218
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]
 

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]
 

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 = 0.91

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

Input:

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

Output:

-1/3/a/x^3+b/a^2/x+4/a^2*c*(1/8*(b*(-4*a*c+b^2)^(1/2)+2*a*c-b^2)/(-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))-1/8*(b*(-4*a*c+b^2)^(1/2)-2*a*c+b^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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1622 vs. \(2 (160) = 320\).

Time = 0.12 (sec) , antiderivative size = 1622, normalized size of antiderivative = 8.28 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 4.71 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{7} c^{2} - 128 a^{6} b^{2} c + 16 a^{5} b^{4}\right ) + t^{2} \left (- 80 a^{3} b c^{3} + 100 a^{2} b^{3} c^{2} - 36 a b^{5} c + 4 b^{7}\right ) + c^{5}, \left ( t \mapsto t \log {\left (x + \frac {- 96 t^{3} a^{7} b c^{2} + 56 t^{3} a^{6} b^{3} c - 8 t^{3} a^{5} b^{5} - 4 t a^{4} c^{4} + 32 t a^{3} b^{2} c^{3} - 40 t a^{2} b^{4} c^{2} + 16 t a b^{6} c - 2 t b^{8}}{a^{2} c^{5} - 3 a b^{2} c^{4} + b^{4} c^{3}} \right )} \right )\right )} + \frac {- a + 3 b x^{2}}{3 a^{2} x^{3}} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

integrate((b*c*x^2 + b^2 - a*c)/(c*x^4 + b*x^2 + a), x)/a^2 + 1/3*(3*b*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. 1640 vs. \(2 (160) = 320\).

Time = 0.74 (sec) , antiderivative size = 1640, normalized size of antiderivative = 8.37 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 4160, normalized size of antiderivative = 21.22 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 810, normalized size of antiderivative = 4.13 \[ \int \frac {1}{x^4 \left (a+b x^2+c x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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