\(\int \frac {1}{a-b x^2+c x^4} \, dx\) [824]

Optimal result

Integrand size = 15, antiderivative size = 150 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=\frac {\sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {-b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1406, 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.

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

\(\Big \downarrow \) 1406

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 1406

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[c/q   Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q   I 
nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c 
, 0] && PosQ[b^2 - 4*a*c]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.27

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}-\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c -\textit {\_R} b}\right )}{2}\)	\(40\)
default	\(4 c \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\)	\(117\)

Input:

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

Output:

1/2*sum(1/(2*_R^3*c-_R*b)*ln(x-_R),_R=RootOf(_Z^4*c-_Z^2*b+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (114) = 228\).

Time = 0.11 (sec) , antiderivative size = 605, normalized size of antiderivative = 4.03 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c - \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x + \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (2 \, c x - \sqrt {\frac {1}{2}} {\left (b^{2} - 4 \, a c + \frac {a b^{3} - 4 \, a^{2} b c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c^{2} - 128 a^{2} b^{2} c + 16 a b^{4}\right ) + t^{2} \cdot \left (16 a b c - 4 b^{3}\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {- 32 t^{3} a^{2} b c + 8 t^{3} a b^{3} + 4 t a c - 2 t b^{2}}{c} \right )} \right )\right )} \] Input:

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

Output:

RootSum(_t**4*(256*a**3*c**2 - 128*a**2*b**2*c + 16*a*b**4) + _t**2*(16*a* 
b*c - 4*b**3) + c, Lambda(_t, _t*log(x + (-32*_t**3*a**2*b*c + 8*_t**3*a*b 
**3 + 4*_t*a*c - 2*_t*b**2)/c)))
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (114) = 228\).

Time = 0.36 (sec) , antiderivative size = 1050, normalized size of antiderivative = 7.00 \[ \int \frac {1}{a-b x^2+c x^4} \, dx =\text {Too large to display} \] Input:

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

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 763, normalized size of antiderivative = 5.09 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=-\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}+b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {\frac {b^3+\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^4\,x\,1{}\mathrm {i}-b\,x\,\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}\,1{}\mathrm {i}+a^2\,c^2\,x\,16{}\mathrm {i}-a\,b^2\,c\,x\,8{}\mathrm {i}}{4\,a\,b^4\,\sqrt {-\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}+64\,a^3\,c^2\,\sqrt {-\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}-32\,a^2\,b^2\,c\,\sqrt {-\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}}\right )\,\sqrt {-\frac {\sqrt {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-b^3+4\,a\,b\,c}{128\,a^3\,c^2-64\,a^2\,b^2\,c+8\,a\,b^4}}\,2{}\mathrm {i} \] Input:

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

Output:

- atan((b^4*x*1i + b*x*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1 
/2)*1i + a^2*c^2*x*16i - a*b^2*c*x*8i)/(4*a*b^4*((b^3 + (b^6 - 64*a^3*c^3 
+ 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 6 
4*a^2*b^2*c))^(1/2) + 64*a^3*c^2*((b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^ 
2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^( 
1/2) - 32*a^2*b^2*c*((b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4* 
c)^(1/2) - 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)))*((b^3 
+ (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(8*a*b 
^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)*2i - atan((b^4*x*1i - b*x*(b^6 - 6 
4*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2)*1i + a^2*c^2*x*16i - a*b^2* 
c*x*8i)/(4*a*b^4*(-((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) 
 - b^3 + 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2) + 64*a^3*c 
^2*(-((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b 
*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2) - 32*a^2*b^2*c*(-((b^6 - 
 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(8*a*b^4 
 + 128*a^3*c^2 - 64*a^2*b^2*c))^(1/2)))*(-((b^6 - 64*a^3*c^3 + 48*a^2*b^2* 
c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(8*a*b^4 + 128*a^3*c^2 - 64*a^2*b 
^2*c))^(1/2)*2i
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.29 \[ \int \frac {1}{a-b x^2+c x^4} \, dx=\frac {-2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b -4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a +2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b +4 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a +\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b -\sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a}{4 a \left (4 a c -b^{2}\right )} \] Input:

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

Output:

( - 2*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))*b - 4*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*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*s 
qrt(c)*sqrt(a) + b) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) - b))*b + 4*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 + 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 - 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*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b)*log( - sqrt(2*s 
qrt(c)*sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*a + 2*sqrt(c)*sqrt(2*sqrt( 
c)*sqrt(a) + b)*log(sqrt(2*sqrt(c)*sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2 
)*a)/(4*a*(4*a*c - b**2))