\(\int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx\) [109]

Optimal result

Integrand size = 23, antiderivative size = 136 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=-\frac {(A+a B) \arctan \left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {(A+a B) \arctan \left (\sqrt {3}+\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {(A-a B) \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}} \]

[Out]

1/2*(B*a+A)*arctan(-3^(1/2)+2*x/a^(1/2))/a^(3/2)+1/2*(B*a+A)*arctan(3^(1/2)+2*x/a^(1/2))/a^(3/2)-1/12*(-B*a+A)
*ln(a+x^2-x*3^(1/2)*a^(1/2))/a^(3/2)*3^(1/2)+1/12*(-B*a+A)*ln(a+x^2+x*3^(1/2)*a^(1/2))/a^(3/2)*3^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1183, 648, 631, 210, 642} \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=-\frac {(a B+A) \arctan \left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {(a B+A) \arctan \left (\frac {2 x}{\sqrt {a}}+\sqrt {3}\right )}{2 a^{3/2}}-\frac {(A-a B) \log \left (-\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \log \left (\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {3} a^{3/2}} \]

[In]

Int[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]

[Out]

-1/2*((A + a*B)*ArcTan[Sqrt[3] - (2*x)/Sqrt[a]])/a^(3/2) + ((A + a*B)*ArcTan[Sqrt[3] + (2*x)/Sqrt[a]])/(2*a^(3
/2)) - ((A - a*B)*Log[a - Sqrt[3]*Sqrt[a]*x + x^2])/(4*Sqrt[3]*a^(3/2)) + ((A - a*B)*Log[a + Sqrt[3]*Sqrt[a]*x
+ x^2])/(4*Sqrt[3]*a^(3/2))

Rule 210

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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1183

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]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {3} \sqrt {a} A-(A-a B) x}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx}{2 \sqrt {3} a^{3/2}}+\frac {\int \frac {\sqrt {3} \sqrt {a} A+(A-a B) x}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx}{2 \sqrt {3} a^{3/2}} \\ & = -\frac {(A-a B) \int \frac {-\sqrt {3} \sqrt {a}+2 x}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \int \frac {\sqrt {3} \sqrt {a}+2 x}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 \sqrt {3} a^{3/2}}+\frac {(A+a B) \int \frac {1}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 a}+\frac {(A+a B) \int \frac {1}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 a} \\ & = -\frac {(A-a B) \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A+a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 x}{\sqrt {3} \sqrt {a}}\right )}{2 \sqrt {3} a^{3/2}}-\frac {(A+a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 x}{\sqrt {3} \sqrt {a}}\right )}{2 \sqrt {3} a^{3/2}} \\ & = -\frac {(A+a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {(A+a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {(A-a B) \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {3} a^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\frac {\sqrt [4]{-1} \left (\frac {\left (-2 i A+\left (-i+\sqrt {3}\right ) a B\right ) \arctan \left (\frac {(1+i) x}{\sqrt {-i+\sqrt {3}} \sqrt {a}}\right )}{\sqrt {-i+\sqrt {3}}}-\frac {\left (2 i A+\left (i+\sqrt {3}\right ) a B\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {i+\sqrt {3}} \sqrt {a}}\right )}{\sqrt {i+\sqrt {3}}}\right )}{\sqrt {6} a^{3/2}} \]

[In]

Integrate[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]

[Out]

((-1)^(1/4)*((((-2*I)*A + (-I + Sqrt[3])*a*B)*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*Sqrt[a])])/Sqrt[-I + Sqrt
[3]] - (((2*I)*A + (I + Sqrt[3])*a*B)*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*Sqrt[a])])/Sqrt[I + Sqrt[3]]))/(S
qrt[6]*a^(3/2))

Maple [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.34

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{2}+a^{2}\right )}{\sum }\frac {\left (B \,\textit {\_R}^{2}+A \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-\textit {\_R} a}\right )}{2}\)	\(46\)
default	\(\frac {\frac {\left (-B \sqrt {3}\, a^{2}+A \sqrt {3}\, a \right ) \ln \left (a +x^{2}+x \sqrt {3}\, \sqrt {a}\right )}{2}+\frac {2 \left (3 A \,a^{\frac {3}{2}}-\frac {\left (-B \sqrt {3}\, a^{2}+A \sqrt {3}\, a \right ) \sqrt {3}\, \sqrt {a}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{\sqrt {a}}}{6 a^{\frac {5}{2}}}+\frac {-\frac {\left (-B \sqrt {3}\, a^{2}+A \sqrt {3}\, a \right ) \ln \left (x \sqrt {3}\, \sqrt {a}-x^{2}-a \right )}{2}+\frac {2 \left (-3 A \,a^{\frac {3}{2}}+\frac {\left (-B \sqrt {3}\, a^{2}+A \sqrt {3}\, a \right ) \sqrt {3}\, \sqrt {a}}{2}\right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {a}-2 x}{\sqrt {a}}\right )}{\sqrt {a}}}{6 a^{\frac {5}{2}}}\)	\(186\)

[In]

int((B*x^2+A)/(x^4-a*x^2+a^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (104) = 208\).

Time = 0.29 (sec) , antiderivative size = 903, normalized size of antiderivative = 6.64 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{6}} \sqrt {-\frac {B^{2} a^{2} + 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}} \log \left (2 \, {\left (B^{4} a^{4} + A B^{3} a^{3} - A^{3} B a - A^{4}\right )} x + 3 \, \sqrt {\frac {1}{6}} {\left (A B^{2} a^{4} - A^{3} a^{2} - \sqrt {\frac {1}{3}} {\left (2 \, B a^{6} + A a^{5}\right )} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}}\right )} \sqrt {-\frac {B^{2} a^{2} + 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{6}} \sqrt {-\frac {B^{2} a^{2} + 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}} \log \left (2 \, {\left (B^{4} a^{4} + A B^{3} a^{3} - A^{3} B a - A^{4}\right )} x - 3 \, \sqrt {\frac {1}{6}} {\left (A B^{2} a^{4} - A^{3} a^{2} - \sqrt {\frac {1}{3}} {\left (2 \, B a^{6} + A a^{5}\right )} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}}\right )} \sqrt {-\frac {B^{2} a^{2} + 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{6}} \sqrt {-\frac {B^{2} a^{2} - 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}} \log \left (2 \, {\left (B^{4} a^{4} + A B^{3} a^{3} - A^{3} B a - A^{4}\right )} x + 3 \, \sqrt {\frac {1}{6}} {\left (A B^{2} a^{4} - A^{3} a^{2} + \sqrt {\frac {1}{3}} {\left (2 \, B a^{6} + A a^{5}\right )} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}}\right )} \sqrt {-\frac {B^{2} a^{2} - 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{6}} \sqrt {-\frac {B^{2} a^{2} - 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}} \log \left (2 \, {\left (B^{4} a^{4} + A B^{3} a^{3} - A^{3} B a - A^{4}\right )} x - 3 \, \sqrt {\frac {1}{6}} {\left (A B^{2} a^{4} - A^{3} a^{2} + \sqrt {\frac {1}{3}} {\left (2 \, B a^{6} + A a^{5}\right )} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}}\right )} \sqrt {-\frac {B^{2} a^{2} - 3 \, \sqrt {\frac {1}{3}} a^{3} \sqrt {-\frac {B^{4} a^{4} - 2 \, A^{2} B^{2} a^{2} + A^{4}}{a^{6}}} + 4 \, A B a + A^{2}}{a^{3}}}\right ) \]

[In]

integrate((B*x^2+A)/(x^4-a*x^2+a^2),x, algorithm="fricas")

[Out]

1/2*sqrt(1/6)*sqrt(-(B^2*a^2 + 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3
)*log(2*(B^4*a^4 + A*B^3*a^3 - A^3*B*a - A^4)*x + 3*sqrt(1/6)*(A*B^2*a^4 - A^3*a^2 - sqrt(1/3)*(2*B*a^6 + A*a^
5)*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt(-(B^2*a^2 + 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2
+ A^4)/a^6) + 4*A*B*a + A^2)/a^3)) - 1/2*sqrt(1/6)*sqrt(-(B^2*a^2 + 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^
2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3)*log(2*(B^4*a^4 + A*B^3*a^3 - A^3*B*a - A^4)*x - 3*sqrt(1/6)*(A*B^2*a^4
- A^3*a^2 - sqrt(1/3)*(2*B*a^6 + A*a^5)*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt(-(B^2*a^2 + 3*sqrt(1
/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3)) + 1/2*sqrt(1/6)*sqrt(-(B^2*a^2 - 3*s
qrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3)*log(2*(B^4*a^4 + A*B^3*a^3 - A^3
*B*a - A^4)*x + 3*sqrt(1/6)*(A*B^2*a^4 - A^3*a^2 + sqrt(1/3)*(2*B*a^6 + A*a^5)*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2
+ A^4)/a^6))*sqrt(-(B^2*a^2 - 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3)
) - 1/2*sqrt(1/6)*sqrt(-(B^2*a^2 - 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) + 4*A*B*a + A^2)
/a^3)*log(2*(B^4*a^4 + A*B^3*a^3 - A^3*B*a - A^4)*x - 3*sqrt(1/6)*(A*B^2*a^4 - A^3*a^2 + sqrt(1/3)*(2*B*a^6 +
A*a^5)*sqrt(-(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt(-(B^2*a^2 - 3*sqrt(1/3)*a^3*sqrt(-(B^4*a^4 - 2*A^2*B^2
*a^2 + A^4)/a^6) + 4*A*B*a + A^2)/a^3))

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\operatorname {RootSum} {\left (144 t^{4} a^{6} + t^{2} \cdot \left (12 A^{2} a^{3} + 48 A B a^{4} + 12 B^{2} a^{5}\right ) + A^{4} + 2 A^{3} B a + 3 A^{2} B^{2} a^{2} + 2 A B^{3} a^{3} + B^{4} a^{4}, \left ( t \mapsto t \log {\left (x + \frac {24 t^{3} A a^{5} + 48 t^{3} B a^{6} - 2 t A^{3} a^{2} + 6 t A^{2} B a^{3} + 12 t A B^{2} a^{4} + 2 t B^{3} a^{5}}{- A^{4} - A^{3} B a + A B^{3} a^{3} + B^{4} a^{4}} \right )} \right )\right )} \]

[In]

integrate((B*x**2+A)/(x**4-a*x**2+a**2),x)

[Out]

RootSum(144*_t**4*a**6 + _t**2*(12*A**2*a**3 + 48*A*B*a**4 + 12*B**2*a**5) + A**4 + 2*A**3*B*a + 3*A**2*B**2*a
**2 + 2*A*B**3*a**3 + B**4*a**4, Lambda(_t, _t*log(x + (24*_t**3*A*a**5 + 48*_t**3*B*a**6 - 2*_t*A**3*a**2 + 6
*_t*A**2*B*a**3 + 12*_t*A*B**2*a**4 + 2*_t*B**3*a**5)/(-A**4 - A**3*B*a + A*B**3*a**3 + B**4*a**4))))

Maxima [F]

\[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\int { \frac {B x^{2} + A}{x^{4} - a x^{2} + a^{2}} \,d x } \]

[In]

integrate((B*x^2+A)/(x^4-a*x^2+a^2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/(x^4 - a*x^2 + a^2), x)

Giac [C] (verification not implemented)

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

Time = 10.84 (sec) , antiderivative size = 4293, normalized size of antiderivative = 31.57 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\text {Too large to display} \]

[In]

integrate((B*x^2+A)/(x^4-a*x^2+a^2),x, algorithm="giac")

[Out]

1/6*sqrt(3)*(3*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(arccos
(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a)))) - sqrt(3)*B*a^2*abs(a)^(3/2)*cosh(1/2*imag_part(ar
ccos(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3 - 9*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_
part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a
))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 3*sqrt(3)*B*a^2*abs(a)^(3/2)*cosh(1/2*imag_part(arccos(1/2*a/
abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 9*sqrt(3)*B
*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2
*real_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 - 3*sqrt(3)*B*a^2*abs(a)^(3/2)*c
osh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(
1/2*a/abs(a))))^2 - 3*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(
arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + sqrt(3)*B*a^2*abs(a)^(3/2)*sin(1/2*real_p
art(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + B*a^3*sqrt(abs(a))*cos(1/2*real_par
t(arccos(1/2*a/abs(a))))^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - 3*B*a^3*sqrt(abs(a))*cos(1/2*real_par
t(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a))))^
2 - 3*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^
2*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 9*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))*co
sh(1/2*imag_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a))))^2*sinh(1/2*imag_part(arccos
(1/2*a/abs(a)))) + 3*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^3*cosh(1/2*imag_part(arccos(1
/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 - 9*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2
*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(1/2*a/abs(a))))^2*sinh(1/2*ima
g_part(arccos(1/2*a/abs(a))))^2 - B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_
part(arccos(1/2*a/abs(a))))^3 + 3*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))*sin(1/2*real_par
t(arccos(1/2*a/abs(a))))^2*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + sqrt(3)*A*a^2*sqrt(abs(a))*cosh(1/2*i
mag_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(1/2*a/abs(a)))) - sqrt(3)*A*a^2*sqrt(abs(a))*sin(1/2*
real_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + A*a*abs(a)^(3/2)*cos(1/2*real_par
t(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a)))) - A*a*abs(a)^(3/2)*cos(1/2*real_part(arccos
(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))))*arctan((sqrt(abs(a))*cos(1/2*arccos(1/2*a/abs(a)))
+ x)/(sqrt(abs(a))*sin(1/2*arccos(1/2*a/abs(a)))))/a^4 + 1/12*sqrt(3)*(sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*rea
l_part(arccos(1/2*a/abs(a))))^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - 3*sqrt(3)*B*a^2*abs(a)^(3/2)*cos
(1/2*real_part(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2
*a/abs(a))))^2 - 3*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))^3*cosh(1/2*imag_part(ar
ccos(1/2*a/abs(a))))^2*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 9*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_p
art(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a)))
)^2*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 3*sqrt(3)*B*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/ab
s(a))))^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 - 9*sqrt(3)*B*
a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2*re
al_part(arccos(1/2*a/abs(a))))^2*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 - sqrt(3)*B*a^2*abs(a)^(3/2)*cos(
1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + 3*sqrt(3)*B*a^2*abs(a)^(3
/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(1/2*a/abs(a))))^2*sinh(1/2*imag_part(arc
cos(1/2*a/abs(a))))^3 - 3*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(arc
cos(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a)))) + B*a^3*sqrt(abs(a))*cosh(1/2*imag_part(arccos(
1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3 + 9*B*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1
/2*a/abs(a))))^2*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a))))*sinh(1/2
*imag_part(arccos(1/2*a/abs(a)))) - 3*B*a^3*sqrt(abs(a))*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^2*sin(1/2*r
eal_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a)))) - 9*B*a^3*sqrt(abs(a))*cos(1/2*rea
l_part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(1/2*a/abs(a
))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 + 3*B*a^3*sqrt(abs(a))*cosh(1/2*imag_part(arccos(1/2*a/abs(a)
)))*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 + 3*B*a^3*sqrt(abs(
a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*sin(1/2*real_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arc
cos(1/2*a/abs(a))))^3 - B*a^3*sqrt(abs(a))*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arcco
s(1/2*a/abs(a))))^3 + sqrt(3)*A*a^2*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(a
rccos(1/2*a/abs(a)))) - sqrt(3)*A*a^2*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part
(arccos(1/2*a/abs(a)))) - A*a*abs(a)^(3/2)*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))*sin(1/2*real_part(arccos(
1/2*a/abs(a)))) + A*a*abs(a)^(3/2)*sin(1/2*real_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/ab
s(a)))))*log(2*x*sqrt(abs(a))*cos(1/2*arccos(1/2*a/abs(a))) + x^2 + abs(a))/a^4 + 1/48*sqrt(3)*(sqrt(3)*sqrt(1
4*a^2 + 13*a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 - 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*
B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 3*sqrt(3
)*sqrt(2*a^2 - a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn(cos(1/2*arccos(1/
2*a/abs(a))))) + 3*sqrt(3)*sqrt(14*a^2 + 13*a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real
_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2 + 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arc
cos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2 - 6*sqrt(3)*sqrt(2*a^2 - a*abs(a))*B*a^5*
imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(
1/2*arccos(1/2*a/abs(a))))) + 3*sqrt(3)*sqrt(14*a^2 - 13*a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/ab
s(a)))))^2*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) - 6*sqrt(3)*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sg
n(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1
/2*a/abs(a))))) + 3*sqrt(3)*sqrt(2*a^2 - a*abs(a))*B*a^5*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_pa
rt(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + sqrt(3)*sqrt(14*a^2 - 13*a*abs(a))*B*a^5*real_part(sgn(sin(1/2*arcc
os(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 - a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*imag_part
(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + sqrt(14*a^2 - 13*a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/a
bs(a)))))^3 - 3*sqrt(14*a^2 + 13*a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn
(cos(1/2*arccos(1/2*a/abs(a))))) + 18*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))
))*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) - sqrt(14*a^2 +
13*a*abs(a))*B*a^5*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part
(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 18*sqrt(2*a^2 - a*abs(a
))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part
(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 9*sqrt(2*a^2 + a*abs(a))*B*a^5*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a
)))))^2*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 9*sqrt(2*a^2 - a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*a
rccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + 3*sqrt(14*a^2 - 13*a*abs(a))*B*a^5*im
ag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + 4*sqrt(3)*sqrt(2
*a^2 + a*abs(a))*A*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) + 4*sqrt(3)*sqrt(2*a^2 - a*abs(a))*A*a^4*
real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) - 4*sqrt(2*a^2 - a*abs(a))*A*a^4*imag_part(sgn(sin(1/2*arccos(1/
2*a/abs(a))))) + 4*sqrt(2*a^2 + a*abs(a))*A*a^4*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))))*arctan(-1/2*(sq
rt(a/abs(a) + 2)*sqrt(abs(a))*sgn(cos(1/2*arccos(1/2*a/abs(a)))) - 2*x)/(sqrt(-1/4*a/abs(a) + 1/2)*sqrt(abs(a)
)*sgn(sin(1/2*arccos(1/2*a/abs(a))))))/(a^5*abs(a)^(3/2)) + 1/96*sqrt(3)*(3*sqrt(3)*sqrt(2*a^2 - a*abs(a))*B*a
^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + sqrt(3)*sqr
t(14*a^2 - 13*a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^3 + 3*sqrt(3)*sqrt(14*a^2 + 13*a*a
bs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) + 6
*sqrt(3)*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*imag_part(sgn(sin(1/2*arcc
os(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) + sqrt(3)*sqrt(14*a^2 + 13*a*abs(a))*B*a^5*r
eal_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 + 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*
arccos(1/2*a/abs(a)))))^2*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 6*sqrt(3)*sqrt(2*a^2 - a*abs(a))*B*a
^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(s
in(1/2*arccos(1/2*a/abs(a))))) + 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*B*a^5*real_part(sgn(cos(1/2*arccos(1/2*a/abs
(a)))))^2*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 3*sqrt(3)*sqrt(2*a^2 - a*abs(a))*B*a^5*imag_part(sgn
(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + 3*sqrt(3)*sqrt(14*a^2 - 13*
a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2
- sqrt(14*a^2 + 13*a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 + a*abs(a))*
B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 9*sqrt(2
*a^2 - a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn(cos(1/2*arccos(1/2*a/abs(
a))))) + 3*sqrt(14*a^2 + 13*a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/
2*arccos(1/2*a/abs(a)))))^2 + 9*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*rea
l_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^2 + 18*sqrt(2*a^2 - a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/
2*a/abs(a)))))*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) - 3
*sqrt(14*a^2 - 13*a*abs(a))*B*a^5*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2*real_part(sgn(sin(1/2*arccos
(1/2*a/abs(a))))) + 18*sqrt(2*a^2 + a*abs(a))*B*a^5*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sg
n(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 9*sqrt(2*a^2 - a*abs(a))*B*a
^5*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 - sqrt(14*a^2
- 13*a*abs(a))*B*a^5*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^3 + 4*sqrt(3)*sqrt(2*a^2 - a*abs(a))*A*a^4
*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 4*sqrt(3)*sqrt(2*a^2 + a*abs(a))*A*a^4*real_part(sgn(cos(1/2*
arccos(1/2*a/abs(a))))) + 4*sqrt(2*a^2 + a*abs(a))*A*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) + 4*sqr
t(2*a^2 - a*abs(a))*A*a^4*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))))*log(-x*sqrt(a/abs(a) + 2)*sqrt(abs(a)
)*sgn(cos(1/2*arccos(1/2*a/abs(a)))) + x^2 + abs(a))/(a^5*abs(a)^(3/2))

Mupad [B] (verification not implemented)

Time = 13.84 (sec) , antiderivative size = 1007, normalized size of antiderivative = 7.40 \[ \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx=\mathrm {atan}\left (\frac {A^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}-\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}+\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}\,6{}\mathrm {i}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2+\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a-\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}+\frac {2\,\sqrt {3}\,A^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}-\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}+\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2+\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a-\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}-\frac {B^2\,a^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}-\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}+\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}\,6{}\mathrm {i}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2+\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a-\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}-\frac {2\,\sqrt {3}\,B^2\,a^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}-\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}+\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2+\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a-\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {A^2+B^2\,a^2+4\,A\,B\,a+\sqrt {3}\,A^2\,1{}\mathrm {i}-\sqrt {3}\,B^2\,a^2\,1{}\mathrm {i}}{24\,a^3}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {A^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}+\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}-\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}\,6{}\mathrm {i}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2-\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a+\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}-\frac {2\,\sqrt {3}\,A^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}+\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}-\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2-\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a+\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}-\frac {B^2\,a^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}+\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}-\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}\,6{}\mathrm {i}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2-\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a+\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}+\frac {2\,\sqrt {3}\,B^2\,a^2\,x\,\sqrt {-\frac {A^2}{24\,a^3}-\frac {B^2}{24\,a}-\frac {A\,B}{6\,a^2}+\frac {\sqrt {3}\,A^2\,1{}\mathrm {i}}{24\,a^3}-\frac {\sqrt {3}\,B^2\,1{}\mathrm {i}}{24\,a}}}{2\,A^2\,B+\frac {A^3}{a}-2\,B^3\,a^2-\frac {\sqrt {3}\,A^3\,1{}\mathrm {i}}{a}-A\,B^2\,a+\sqrt {3}\,A\,B^2\,a\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {A^2+B^2\,a^2+4\,A\,B\,a-\sqrt {3}\,A^2\,1{}\mathrm {i}+\sqrt {3}\,B^2\,a^2\,1{}\mathrm {i}}{24\,a^3}}\,2{}\mathrm {i} \]

[In]

int((A + B*x^2)/(a^2 - a*x^2 + x^4),x)

[Out]

atan((A^2*x*((3^(1/2)*B^2*1i)/(24*a) - B^2/(24*a) - (3^(1/2)*A^2*1i)/(24*a^3) - A^2/(24*a^3) - (A*B)/(6*a^2))^
(1/2)*6i)/(2*A^2*B + A^3/a - 2*B^3*a^2 + (3^(1/2)*A^3*1i)/a - A*B^2*a - 3^(1/2)*A*B^2*a*1i) + (2*3^(1/2)*A^2*x
*((3^(1/2)*B^2*1i)/(24*a) - B^2/(24*a) - (3^(1/2)*A^2*1i)/(24*a^3) - A^2/(24*a^3) - (A*B)/(6*a^2))^(1/2))/(2*A
^2*B + A^3/a - 2*B^3*a^2 + (3^(1/2)*A^3*1i)/a - A*B^2*a - 3^(1/2)*A*B^2*a*1i) - (B^2*a^2*x*((3^(1/2)*B^2*1i)/(
24*a) - B^2/(24*a) - (3^(1/2)*A^2*1i)/(24*a^3) - A^2/(24*a^3) - (A*B)/(6*a^2))^(1/2)*6i)/(2*A^2*B + A^3/a - 2*
B^3*a^2 + (3^(1/2)*A^3*1i)/a - A*B^2*a - 3^(1/2)*A*B^2*a*1i) - (2*3^(1/2)*B^2*a^2*x*((3^(1/2)*B^2*1i)/(24*a) -
B^2/(24*a) - (3^(1/2)*A^2*1i)/(24*a^3) - A^2/(24*a^3) - (A*B)/(6*a^2))^(1/2))/(2*A^2*B + A^3/a - 2*B^3*a^2 +
(3^(1/2)*A^3*1i)/a - A*B^2*a - 3^(1/2)*A*B^2*a*1i))*(-(3^(1/2)*A^2*1i + A^2 + B^2*a^2 - 3^(1/2)*B^2*a^2*1i + 4
*A*B*a)/(24*a^3))^(1/2)*2i + atan((A^2*x*((3^(1/2)*A^2*1i)/(24*a^3) - B^2/(24*a) - A^2/(24*a^3) - (3^(1/2)*B^2
*1i)/(24*a) - (A*B)/(6*a^2))^(1/2)*6i)/(2*A^2*B + A^3/a - 2*B^3*a^2 - (3^(1/2)*A^3*1i)/a - A*B^2*a + 3^(1/2)*A
*B^2*a*1i) - (2*3^(1/2)*A^2*x*((3^(1/2)*A^2*1i)/(24*a^3) - B^2/(24*a) - A^2/(24*a^3) - (3^(1/2)*B^2*1i)/(24*a)
- (A*B)/(6*a^2))^(1/2))/(2*A^2*B + A^3/a - 2*B^3*a^2 - (3^(1/2)*A^3*1i)/a - A*B^2*a + 3^(1/2)*A*B^2*a*1i) - (
B^2*a^2*x*((3^(1/2)*A^2*1i)/(24*a^3) - B^2/(24*a) - A^2/(24*a^3) - (3^(1/2)*B^2*1i)/(24*a) - (A*B)/(6*a^2))^(1
/2)*6i)/(2*A^2*B + A^3/a - 2*B^3*a^2 - (3^(1/2)*A^3*1i)/a - A*B^2*a + 3^(1/2)*A*B^2*a*1i) + (2*3^(1/2)*B^2*a^2
*x*((3^(1/2)*A^2*1i)/(24*a^3) - B^2/(24*a) - A^2/(24*a^3) - (3^(1/2)*B^2*1i)/(24*a) - (A*B)/(6*a^2))^(1/2))/(2
*A^2*B + A^3/a - 2*B^3*a^2 - (3^(1/2)*A^3*1i)/a - A*B^2*a + 3^(1/2)*A*B^2*a*1i))*(-(A^2 - 3^(1/2)*A^2*1i + B^2
*a^2 + 3^(1/2)*B^2*a^2*1i + 4*A*B*a)/(24*a^3))^(1/2)*2i