\(\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx\) [106]
Optimal result
Integrand size = 25, antiderivative size = 114 \[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=-\frac {\arctan \left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\arctan \left (\sqrt {3}+\frac {2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {3} \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}}+\frac {\sqrt {3} \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}}
\]
[Out]
1/2*arctan(-3^(1/2)+2*x/a^(1/2))/a^(1/2)+1/2*arctan(3^(1/2)+2*x/a^(1/2))/a^(1/2)-1/4*ln(a+x^2-x*3^(1/2)*a^(1/2
))*3^(1/2)/a^(1/2)+1/4*ln(a+x^2+x*3^(1/2)*a^(1/2))*3^(1/2)/a^(1/2)
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1183, 648, 631, 210, 642}
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=-\frac {\arctan \left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\arctan \left (\frac {2 x}{\sqrt {a}}+\sqrt {3}\right )}{2 \sqrt {a}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {a}}
\]
[In]
Int[(2*a - x^2)/(a^2 - a*x^2 + x^4),x]
[Out]
-1/2*ArcTan[Sqrt[3] - (2*x)/Sqrt[a]]/Sqrt[a] + ArcTan[Sqrt[3] + (2*x)/Sqrt[a]]/(2*Sqrt[a]) - (Sqrt[3]*Log[a -
Sqrt[3]*Sqrt[a]*x + x^2])/(4*Sqrt[a]) + (Sqrt[3]*Log[a + Sqrt[3]*Sqrt[a]*x + x^2])/(4*Sqrt[a])
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 {2 \sqrt {3} a^{3/2}-3 a x}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx}{2 \sqrt {3} a^{3/2}}+\frac {\int \frac {2 \sqrt {3} a^{3/2}+3 a x}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx}{2 \sqrt {3} a^{3/2}} \\ & = \frac {1}{4} \int \frac {1}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx-\frac {\sqrt {3} \int \frac {-\sqrt {3} \sqrt {a}+2 x}{a-\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 \sqrt {a}}+\frac {\sqrt {3} \int \frac {\sqrt {3} \sqrt {a}+2 x}{a+\sqrt {3} \sqrt {a} x+x^2} \, dx}{4 \sqrt {a}} \\ & = -\frac {\sqrt {3} \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}}+\frac {\sqrt {3} \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}}+\frac {\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} \sqrt {a}}-\frac {\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} \sqrt {a}} \\ & = -\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {3} \log \left (a-\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}}+\frac {\sqrt {3} \log \left (a+\sqrt {3} \sqrt {a} x+x^2\right )}{4 \sqrt {a}} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=\frac {\sqrt [4]{-1} \left (-\sqrt {i+\sqrt {3}} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {-i+\sqrt {3}} \sqrt {a}}\right )+\sqrt {-i+\sqrt {3}} \left (-3 i+\sqrt {3}\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {i+\sqrt {3}} \sqrt {a}}\right )\right )}{2 \sqrt {6} \sqrt {a}}
\]
[In]
Integrate[(2*a - x^2)/(a^2 - a*x^2 + x^4),x]
[Out]
((-1)^(1/4)*(-(Sqrt[I + Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*Sqrt[a])]) + Sqrt[-I +
Sqrt[3]]*(-3*I + Sqrt[3])*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*Sqrt[a])]))/(2*Sqrt[6]*Sqrt[a])
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.42
| | |
| method | result | size |
| | |
| risch |
\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{2}+a^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+2 a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-\textit {\_R} a}\right )}{2}\) |
\(48\) |
| default |
\(\frac {\frac {\sqrt {3}\, \ln \left (a +x^{2}+x \sqrt {3}\, \sqrt {a}\right )}{2}+\arctan \left (\frac {2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {-\frac {\sqrt {3}\, \ln \left (x \sqrt {3}\, \sqrt {a}-x^{2}-a \right )}{2}-\arctan \left (\frac {\sqrt {3}\, \sqrt {a}-2 x}{\sqrt {a}}\right )}{2 \sqrt {a}}\) |
\(90\) |
| | |
|
|
|
|
[In]
int((-x^2+2*a)/(x^4-a*x^2+a^2),x,method=_RETURNVERBOSE)
[Out]
1/2*sum((-_R^2+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. 219 vs. \(2 (80) = 160\).
Time = 0.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.92
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} + 1}{a}} \log \left (\sqrt {\frac {1}{2}} a \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} + 1}{a}} + x\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} + 1}{a}} \log \left (-\sqrt {\frac {1}{2}} a \sqrt {\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} + 1}{a}} + x\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} - 1}{a}} \log \left (\sqrt {\frac {1}{2}} a \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} - 1}{a}} + x\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} - 1}{a}} \log \left (-\sqrt {\frac {1}{2}} a \sqrt {-\frac {\sqrt {3} a \sqrt {-\frac {1}{a^{2}}} - 1}{a}} + x\right )
\]
[In]
integrate((-x^2+2*a)/(x^4-a*x^2+a^2),x, algorithm="fricas")
[Out]
1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqrt(-1/a^2) + 1)/a)*log(sqrt(1/2)*a*sqrt((sqrt(3)*a*sqrt(-1/a^2) + 1)/a) + x) -
1/2*sqrt(1/2)*sqrt((sqrt(3)*a*sqrt(-1/a^2) + 1)/a)*log(-sqrt(1/2)*a*sqrt((sqrt(3)*a*sqrt(-1/a^2) + 1)/a) + x)
+ 1/2*sqrt(1/2)*sqrt(-(sqrt(3)*a*sqrt(-1/a^2) - 1)/a)*log(sqrt(1/2)*a*sqrt(-(sqrt(3)*a*sqrt(-1/a^2) - 1)/a) +
x) - 1/2*sqrt(1/2)*sqrt(-(sqrt(3)*a*sqrt(-1/a^2) - 1)/a)*log(-sqrt(1/2)*a*sqrt(-(sqrt(3)*a*sqrt(-1/a^2) - 1)/
a) + x)
Sympy [A] (verification not implemented)
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.24
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=- \operatorname {RootSum} {\left (16 t^{4} a^{2} - 4 t^{2} a + 1, \left ( t \mapsto t \log {\left (- 2 t a + x \right )} \right )\right )}
\]
[In]
integrate((-x**2+2*a)/(x**4-a*x**2+a**2),x)
[Out]
-RootSum(16*_t**4*a**2 - 4*_t**2*a + 1, Lambda(_t, _t*log(-2*_t*a + x)))
Maxima [F]
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=\int { -\frac {x^{2} - 2 \, a}{x^{4} - a x^{2} + a^{2}} \,d x }
\]
[In]
integrate((-x^2+2*a)/(x^4-a*x^2+a^2),x, algorithm="maxima")
[Out]
-integrate((x^2 - 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 = 8.56 (sec) , antiderivative size = 4217, normalized size of antiderivative = 36.99
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=\text {Too large to display}
\]
[In]
integrate((-x^2+2*a)/(x^4-a*x^2+a^2),x, algorithm="giac")
[Out]
-1/6*sqrt(3)*(3*sqrt(3)*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)*a^2*abs(a)^(3/2)*cosh(1/2*imag_part(arcco
s(1/2*a/abs(a))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a))))^3 - 9*sqrt(3)*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)*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)*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_par
t(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^2 - 3*sqrt(3)*a^2*abs(a)^(3/2)*cosh(1/2*ima
g_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)*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)*a^2*abs(a)^(3/2)*sin(1/2*real_part(arccos(1/2*
a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/ab
s(a))))^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - 3*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))))^3*sin(1/2*real_part(arccos(1/2*a/abs(a))))^2 - 3*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*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))))^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*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*i
mag_part(arccos(1/2*a/abs(a))))^2 - 9*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*imag_part(arccos(1/2*a/abs(a))
))^2 - 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*a^3*sqrt(abs(a))*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(arccos(1/2*a/abs(a))))^3 - 2*sqrt(3)*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)))) + 2*sqrt(3)*a^3*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)))) - 2*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)))) + 2*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))*sinh(1/2*i
mag_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)*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))
^3*cosh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - 3*sqrt(3)*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)*a^2*
abs(a)^(3/2)*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*i
mag_part(arccos(1/2*a/abs(a)))) + 9*sqrt(3)*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))))^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)*a^2*abs(a)^(3/2)*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*sqrt(3)*a^2*abs(a)^(3/2)*cos(1/2*real_part(arcc
os(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*imag_part(arccos(1/2*a/abs(a))))^2 - sqrt(3)*a^2*abs(a)^(3/2)*cos(1/2*real_part(arccos(1/2*a/abs(a))))^3*si
nh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 + 3*sqrt(3)*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(arccos(1/2*a/abs(a))))^3 - 3*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))))^3*sin(1/2*real_part(arcco
s(1/2*a/abs(a)))) + 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*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*a^3*sqrt(ab
s(a))*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_par
t(arccos(1/2*a/abs(a)))) - 9*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*cosh(1/2*imag_part(ar
ccos(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*
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*a^3*sqrt(abs(a))*cos(1/2*real_part(arccos(1/2*a/abs(a))))^2*sin(1/2*re
al_part(arccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - a^3*sqrt(abs(a))*sin(1/2*real_par
t(arccos(1/2*a/abs(a))))^3*sinh(1/2*imag_part(arccos(1/2*a/abs(a))))^3 - 2*sqrt(3)*a^3*sqrt(abs(a))*cos(1/2*re
al_part(arccos(1/2*a/abs(a))))*cosh(1/2*imag_part(arccos(1/2*a/abs(a)))) + 2*sqrt(3)*a^3*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)))) + 2*a^2*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)))) - 2*a^2*abs(a)^(3/2)*sin(1/2*real_part(ar
ccos(1/2*a/abs(a))))*sinh(1/2*imag_part(arccos(1/2*a/abs(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(14*a^2 + 13*a*abs(a))*a^4*imag_part(sgn(cos(1/2*arccos(1
/2*a/abs(a)))))^3 - 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*a^4*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))*a^4*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))*a^4*ima
g_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))*a^4*imag_part(sgn(sin(1/2*arccos(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))*a^4*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))*
a^4*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))))) - 6*sqrt(3)*
sqrt(2*a^2 + a*abs(a))*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arccos(1/2*a/ab
s(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) - 3*sqrt(3)*sqrt(2*a^2 - a*abs(a))*a^4*real_part(sgn(co
s(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^2 + sqrt(3)*sqrt(14*a^2 - 13*a*abs
(a))*a^4*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 - a*abs(a))*a^4*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))*a^4*imag_
part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^3 - 3*sqrt(14*a^2 + 13*a*abs(a))*a^4*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))*a^4*imag_part(sgn(co
s(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))*a^4*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 +
a*abs(a))*a^4*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))*a^4*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))*a^4*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))*a^4*imag_p
art(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(14*a^2 - 13*a
*abs(a))*a^4*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 + 8
*sqrt(3)*sqrt(2*a^2 + a*abs(a))*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) + 8*sqrt(3)*sqrt(2*a^2 - a*a
bs(a))*a^4*real_part(sgn(sin(1/2*arccos(1/2*a/abs(a))))) + 8*sqrt(2*a^2 - a*abs(a))*a^4*imag_part(sgn(sin(1/2*
arccos(1/2*a/abs(a))))) - 8*sqrt(2*a^2 + a*abs(a))*a^4*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))))*arctan(-
1/2*(sqrt(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^4*abs(a)^(3/2)) - 1/96*sqrt(3)*(3*sqrt(3)*sqrt(2*a^2 - a*abs(
a))*a^4*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
)*sqrt(14*a^2 - 13*a*abs(a))*a^4*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))^3 + 3*sqrt(3)*sqrt(14*a^2 + 13*
a*abs(a))*a^4*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))*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))*imag_part(sgn(sin(1/2*arcco
s(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))*a^4*real
_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 - 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*a^4*imag_part(sgn(cos(1/2*arcco
s(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))*a^4*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))))) - 3*sqrt(3)*sqrt(2*a^2 + a*abs(a))*a^4*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))*a^4*imag_part(sgn(cos(1/2*ar
ccos(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))*a
^4*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))*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a)))))^3 + 9*sqrt(2*a^2 + a*abs(a))*a^4*imag_part(s
gn(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))*
a^4*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))*a^4*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))*a^4*imag_part(sgn(sin(1/2*arccos(1/2*a/abs(a)))))*real_part(sgn(cos(1/2*arcc
os(1/2*a/abs(a)))))^2 + 18*sqrt(2*a^2 - a*abs(a))*a^4*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
))*a^4*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))*a^4*imag_part(sgn(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))))) + 9*sqrt(2*a^2 - a*abs(a))*a^4*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))*a^4*real_part(s
gn(sin(1/2*arccos(1/2*a/abs(a)))))^3 + 8*sqrt(3)*sqrt(2*a^2 - a*abs(a))*a^4*imag_part(sgn(sin(1/2*arccos(1/2*a
/abs(a))))) + 8*sqrt(3)*sqrt(2*a^2 + a*abs(a))*a^4*real_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) - 8*sqrt(2*a^
2 + a*abs(a))*a^4*imag_part(sgn(cos(1/2*arccos(1/2*a/abs(a))))) - 8*sqrt(2*a^2 - a*abs(a))*a^4*real_part(sgn(s
in(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^4*abs(a)^(3/2))
Mupad [B] (verification not implemented)
Time = 13.44 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.17
\[
\int \frac {2 a-x^2}{a^2-a x^2+x^4} \, dx=-\frac {\sqrt {8}\,\mathrm {atan}\left (x\,\sqrt {\frac {1}{8\,a}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8\,a}}\,1{}\mathrm {i}+\sqrt {3}\,x\,\sqrt {\frac {1}{8\,a}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8\,a}}\right )\,\sqrt {\frac {1+\sqrt {3}\,1{}\mathrm {i}}{a}}\,1{}\mathrm {i}}{4}-\frac {\sqrt {8}\,\mathrm {atan}\left (x\,\sqrt {\frac {1}{8\,a}-\frac {\sqrt {3}\,1{}\mathrm {i}}{8\,a}}\,1{}\mathrm {i}-\sqrt {3}\,x\,\sqrt {\frac {1}{8\,a}-\frac {\sqrt {3}\,1{}\mathrm {i}}{8\,a}}\right )\,\sqrt {-\frac {-1+\sqrt {3}\,1{}\mathrm {i}}{a}}\,1{}\mathrm {i}}{4}
\]
[In]
int((2*a - x^2)/(a^2 - a*x^2 + x^4),x)
[Out]
- (8^(1/2)*atan(x*((3^(1/2)*1i)/(8*a) + 1/(8*a))^(1/2)*1i + 3^(1/2)*x*((3^(1/2)*1i)/(8*a) + 1/(8*a))^(1/2))*((
3^(1/2)*1i + 1)/a)^(1/2)*1i)/4 - (8^(1/2)*atan(x*(1/(8*a) - (3^(1/2)*1i)/(8*a))^(1/2)*1i - 3^(1/2)*x*(1/(8*a)
- (3^(1/2)*1i)/(8*a))^(1/2))*(-(3^(1/2)*1i - 1)/a)^(1/2)*1i)/4