3.1.97 \(\int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx\)
Optimal. Leaf size=136 \[ -\frac {(A-a B) \log \left (-\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {3} a^{3/2}}+\frac {(A-a B) \log \left (\sqrt {3} \sqrt {a} x+a+x^2\right )}{4 \sqrt {3} a^{3/2}}-\frac {(a B+A) \tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {(a B+A) \tan ^{-1}\left (\frac {2 x}{\sqrt {a}}+\sqrt {3}\right )}{2 a^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used =
{1169, 634, 617, 204, 628} \begin {gather*} -\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}}-\frac {(a B+A) \tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {(a B+A) \tan ^{-1}\left (\frac {2 x}{\sqrt {a}}+\sqrt {3}\right )}{2 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]
[Out]
-((A + a*B)*ArcTan[Sqrt[3] - (2*x)/Sqrt[a]])/(2*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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 617
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 628
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 634
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 1169
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*} \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx &=\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) \operatorname {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) \operatorname {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] time = 0.15, size = 130, normalized size = 0.96 \begin {gather*} \frac {\sqrt [4]{-1} \left (\frac {\left (\left (\sqrt {3}-i\right ) a B-2 i A\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {\sqrt {3}-i} \sqrt {a}}\right )}{\sqrt {\sqrt {3}-i}}-\frac {\left (\left (\sqrt {3}+i\right ) a B+2 i A\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {\sqrt {3}+i} \sqrt {a}}\right )}{\sqrt {\sqrt {3}+i}}\right )}{\sqrt {6} a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[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))
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{a^2-a x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]
[Out]
IntegrateAlgebraic[(A + B*x^2)/(a^2 - a*x^2 + x^4), x]
________________________________________________________________________________________
fricas [B] time = 2.41, size = 4551, normalized size = 33.46
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x^2+A)/(x^4-a*x^2+a^2),x, algorithm="fricas")
[Out]
1/4*(4*(1/9)^(1/4)*a^6*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^
4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A
^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4
)/a^6)*arctan((18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*
B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*sqrt(
(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2
*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*
A^2*B^2*a^2 + A^4))*sqrt(((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)*x^2 + 3*sqrt(1/3)*(1/9)^(1
/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^
3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^
4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + (B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 +
2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)
)*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) - 18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*
A*a^10*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A
^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*x*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B
^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A
*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3
*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) + 2*sqrt(1/3)*(B^4*a^10 + 2*A*B^3*a^9 + 3*A^2*B^2*a^8 + 2*A^3*B*a^7
+ A^4*a^6)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2
+ A^4)/a^6) + sqrt(1/3)*(B^6*a^9 + 3*A*B^5*a^8 + 6*A^2*B^4*a^7 + 7*A^3*B^3*a^6 + 6*A^4*B^2*a^5 + 3*A^5*B*a^4 +
A^6*a^3)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))/(B^8*a^8 + 3*A*B^7*a^7 + 5*A^2*B^6*a^6 + 4*A^3*B^5*a^5 -
4*A^5*B^3*a^3 - 5*A^6*B^2*a^2 - 3*A^7*B*a - A^8)) + 4*(1/9)^(1/4)*a^6*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^
2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^
3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)
/a^6)^(3/4)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6)*arctan((18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*sqrt(
(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqr
t(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3
*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2
*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*sqrt(((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^
2 + 2*A^3*B*a + A^4)*x^2 - 3*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^
3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3
*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/
a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) +
(B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 +
2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^
6)^(3/4) - 18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*
a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*x*sqrt(
(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2
*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*
A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) - 2*sqrt(1/3)*(B^4*a
^10 + 2*A*B^3*a^9 + 3*A^2*B^2*a^8 + 2*A^3*B*a^7 + A^4*a^6)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3
*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^6*a^9 + 3*A*B^5*a^8 + 6*A^2*B^4*a^7
+ 7*A^3*B^3*a^6 + 6*A^4*B^2*a^5 + 3*A^5*B*a^4 + A^6*a^3)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))/(B^8*a^8 +
3*A*B^7*a^7 + 5*A^2*B^6*a^6 + 4*A^3*B^5*a^5 - 4*A^5*B^3*a^3 - 5*A^6*B^2*a^2 - 3*A^7*B*a - A^8)) - sqrt(1/3)*(
1/9)^(1/4)*(2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 - (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt
((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2
+ 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a
+ A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)
^(1/4)*log(2*(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)*x^2 + 6*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqr
t((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4
+ 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a
^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + 2*(B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3
+ 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)) + sqrt(1/3)*(1/9)^(1/4)*(2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4
*A^3*B*a + 2*A^4 - (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A
^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^
4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4)*log(2*(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2
*A^3*B*a + A^4)*x^2 - 6*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a
+ A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a
+ 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))
/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + 2*(B
^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)))/(B^4*a^4 + 2
*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x^2+A)/(x^4-a*x^2+a^2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-16,[
2,0]%%%}+%%%{-4,[0,1]%%%},0,%%%{64,[4,0]%%%}+%%%{8,[2,2]%%%}+%%%{16,[2,1]%%%}+%%%{6,[0,2]%%%},0,%%%{-64,[4,2]%
%%}+%%%{-128,[4,1]%%%}+%%%{48,[2,3]%%%}+%%%{16,[2,2]%%%}+%%%{-4,[0,3]%%%},0,%%%{16,[4,4]%%%}+%%%{-64,[4,3]%%%}
+%%%{64,[4,2]%%%}+%%%{8,[2,4]%%%}+%%%{-16,[2,3]%%%}+%%%{1,[0,4]%%%}] at parameters values [71,-96]Warning, cho
osing root of [1,0,%%%{-16,[2,0]%%%}+%%%{-4,[0,1]%%%},0,%%%{64,[4,0]%%%}+%%%{8,[2,2]%%%}+%%%{16,[2,1]%%%}+%%%{
6,[0,2]%%%},0,%%%{-64,[4,2]%%%}+%%%{-128,[4,1]%%%}+%%%{48,[2,3]%%%}+%%%{16,[2,2]%%%}+%%%{-4,[0,3]%%%},0,%%%{16
,[4,4]%%%}+%%%{-64,[4,3]%%%}+%%%{64,[4,2]%%%}+%%%{8,[2,4]%%%}+%%%{-16,[2,3]%%%}+%%%{1,[0,4]%%%}] at parameters
values [72,-72]((64*a^3*sqrt(abs(a))*abs(a)+32*sqrt(3)*a^4*sqrt(abs(a))+32*a^4*sqrt(abs(a)))*A*im(sign(cos(ac
os(a/2/abs(a))/2)))+(64*sqrt(3)*a^5+192*abs(a)*a^4-128*sqrt(3)*a^4*abs(a))*1/2/sqrt(abs(a))*A*im(sign(sin(acos
(a/2/abs(a))/2)))+(-64*sqrt(3)*a^5+192*abs(a)*a^4-128*sqrt(3)*a^4*abs(a))*1/2/sqrt(abs(a))*A*re(sign(cos(acos(
a/2/abs(a))/2)))+(32*sqrt(3)*a^5-64*a^5+32*a^4*abs(a))/sqrt(abs(a))*A*re(sign(sin(acos(a/2/abs(a))/2)))+(-32*a
^6-40*a^5*abs(a)+8*sqrt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^3-1/12
*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^2*im(sign(
sin(acos(a/2/abs(a))/2)))-1/24*(-2880*sqrt(3)*a^6+1728*abs(a)*a^4*sqrt(5*a^2+4*a*abs(a))-2304*sqrt(3)*a^5*abs(
a))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^2*re(sign(cos(acos(a/2/abs(a))/2)))-(-72*a^5*abs(a)+24*sq
rt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^2*re(sign(sin(acos(a/2/abs(
a))/2)))-(-72*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)
))*im(sign(sin(acos(a/2/abs(a))/2)))^2-(-144*a^5*abs(a)+48*sqrt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*
im(sign(cos(acos(a/2/abs(a))/2)))*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))+1/24*(-3
456*sqrt(3)*a^6+3456*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*im(sign(sin(
acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))-(-96*a^6-120*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2+4*a*
abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))^2+1/24*(-3456*sqrt
(3)*a^6+3456*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2
/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))+(-72*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(ab
s(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))^2+1/8*(-320*sqrt(3)*a^6+192*abs(a)
*a^4*sqrt(5*a^2-4*a*abs(a))+256*sqrt(3)*a^5*abs(a))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))^3+1/12*(-
864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))^2*re(sign(cos
(acos(a/2/abs(a))/2)))+(96*a^6-120*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(si
n(acos(a/2/abs(a))/2)))^2*re(sign(sin(acos(a/2/abs(a))/2)))+1/12*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2+4*a*abs(
a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))^2+(-144*a^5*abs(a)+48*
sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(
a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))-1/24*(-2880*sqrt(3)*a^6+1728*abs(a)*a^4*sqrt(5*a^2-4*a*abs(a))+2304
*sqrt(3)*a^5*abs(a))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))^2+1/8*
(-320*sqrt(3)*a^6+192*abs(a)*a^4*sqrt(5*a^2+4*a*abs(a))-256*sqrt(3)*a^5*abs(a))/sqrt(abs(a))*B*re(sign(cos(aco
s(a/2/abs(a))/2)))^3+(-72*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*re(sign(cos(acos(a/
2/abs(a))/2)))^2*re(sign(sin(acos(a/2/abs(a))/2)))-1/12*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt
(abs(a))*B*re(sign(cos(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))^2-(32*a^6-40*a^5*abs(a)+8*sqrt(
3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*re(sign(sin(acos(a/2/abs(a))/2)))^3)/(256*a^4*sqrt(2*a^2+a*abs(a
))*sqrt(3)*abs(a)-256*a^4*sqrt(2*a^2-a*abs(a))*sqrt(3)*abs(a))*ln(x^2-2*sqrt((1+a*1/2/abs(a))/2)*sqrt(abs(a))*
sign(cos(acos(a*1/2/abs(a))/2))*x+sqrt(abs(a))*sqrt(abs(a)))+((64*sqrt(3)*a^5+192*abs(a)*a^4+128*sqrt(3)*a^4*a
bs(a))*1/2/sqrt(abs(a))*A*im(sign(cos(acos(a/2/abs(a))/2)))+(64*a^3*sqrt(abs(a))*abs(a)+32*sqrt(3)*a^4*sqrt(ab
s(a))-32*a^4*sqrt(abs(a)))*A*im(sign(sin(acos(a/2/abs(a))/2)))+(64*a^3*sqrt(abs(a))*abs(a)+32*sqrt(3)*a^4*sqrt
(abs(a))+32*a^4*sqrt(abs(a)))*A*re(sign(cos(acos(a/2/abs(a))/2)))+(64*sqrt(3)*a^5+192*abs(a)*a^4-128*sqrt(3)*a
^4*abs(a))*1/2/sqrt(abs(a))*A*re(sign(sin(acos(a/2/abs(a))/2)))+1/8*(-320*sqrt(3)*a^6+192*abs(a)*a^4*sqrt(5*a^
2+4*a*abs(a))-256*sqrt(3)*a^5*abs(a))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^3+(-72*a^5*abs(a)+24*sq
rt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^2*im(sign(sin(acos(a/2/abs(
a))/2)))+(-96*a^6-120*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/ab
s(a))/2)))^2*re(sign(cos(acos(a/2/abs(a))/2)))-1/12*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs
(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))^2*re(sign(sin(acos(a/2/abs(a))/2)))-1/12*(-864*sqrt(3)*a^6+864*a^5*sq
rt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*im(sign(sin(acos(a/2/abs(a))/2)))^2-1/2
4*(-3456*sqrt(3)*a^6+3456*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*im(sign
(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))-(-144*a^5*abs(a)+48*sqrt(3)*a^5*sqrt(5*a^2-4*a*ab
s(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2
/abs(a))/2)))-1/24*(-2880*sqrt(3)*a^6+1728*abs(a)*a^4*sqrt(5*a^2+4*a*abs(a))-2304*sqrt(3)*a^5*abs(a))/sqrt(abs
(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))^2-(-144*a^5*abs(a)+48*sqrt(3)*a^5*s
qrt(5*a^2+4*a*abs(a)))/sqrt(abs(a))*B*im(sign(cos(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))*re(s
ign(sin(acos(a/2/abs(a))/2)))+1/12*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(co
s(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))^2-(32*a^6-40*a^5*abs(a)+8*sqrt(3)*a^5*sqrt(5*a^2-4*a
*abs(a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))^3-(-72*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs
(a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))^2*re(sign(cos(acos(a/2/abs(a))/2)))+1/24*(-2880*sqrt(3)
*a^6+1728*abs(a)*a^4*sqrt(5*a^2-4*a*abs(a))+2304*sqrt(3)*a^5*abs(a))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a
))/2)))^2*re(sign(sin(acos(a/2/abs(a))/2)))-(-72*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2+4*a*abs(a)))/sqrt(abs(a)
)*B*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))^2+1/24*(-3456*sqrt(3)*a^6+3456*a^5*sqr
t(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(sign(sin(acos(a/2/abs(a))/2)))*re(sign(cos(acos(a/2/abs(a))/2)))*re(sig
n(sin(acos(a/2/abs(a))/2)))+(96*a^6-120*a^5*abs(a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*im(si
gn(sin(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/2/abs(a))/2)))^2-(-32*a^6-40*a^5*abs(a)+8*sqrt(3)*a^5*sqrt(5*a
^2+4*a*abs(a)))/sqrt(abs(a))*B*re(sign(cos(acos(a/2/abs(a))/2)))^3+1/12*(-864*sqrt(3)*a^6+864*a^5*sqrt(5*a^2+4
*a*abs(a)))/sqrt(abs(a))*B*re(sign(cos(acos(a/2/abs(a))/2)))^2*re(sign(sin(acos(a/2/abs(a))/2)))+(-72*a^5*abs(
a)+24*sqrt(3)*a^5*sqrt(5*a^2-4*a*abs(a)))/sqrt(abs(a))*B*re(sign(cos(acos(a/2/abs(a))/2)))*re(sign(sin(acos(a/
2/abs(a))/2)))^2-1/8*(-320*sqrt(3)*a^6+192*abs(a)*a^4*sqrt(5*a^2-4*a*abs(a))+256*sqrt(3)*a^5*abs(a))/sqrt(abs(
a))*B*re(sign(sin(acos(a/2/abs(a))/2)))^3)/(128*a^4*sqrt(2*a^2+a*abs(a))*sqrt(3)*abs(a)-128*a^4*sqrt(2*a^2-a*a
bs(a))*sqrt(3)*abs(a))*atan((x-sign(cos(acos(a*1/2/abs(a))/2))*sqrt((1+a*1/2/abs(a))/2)*sqrt(abs(a)))/sign(sin
(acos(a*1/2/abs(a))/2))/sqrt((1-a*1/2/abs(a))/2)/sqrt(abs(a)))+(-abs(a)*sqrt(abs(a))*A*a*cosh(im(acos(a/2/abs(
a)))/2)*sin(re(acos(a/2/abs(a)))/2)+abs(a)*sqrt(abs(a))*A*a*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a
)))/2)+sqrt(3)*a^2*sqrt(abs(a))*A*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)-sqrt(3)*a^2*sqrt(ab
s(a))*A*cos(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)-3*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)
))/2)^2*cosh(im(acos(a/2/abs(a)))/2)^3*sin(re(acos(a/2/abs(a)))/2)+9*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(
a)))/2)^2*cosh(im(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)-9*a^2*sqrt(a
bs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)^2*cosh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos
(a/2/abs(a)))/2)^2+3*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)*sinh(im(ac
os(a/2/abs(a)))/2)^3+a^2*sqrt(abs(a))*B*a*cosh(im(acos(a/2/abs(a)))/2)^3*sin(re(acos(a/2/abs(a)))/2)^3-3*a^2*s
qrt(abs(a))*B*a*cosh(im(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)+3*a^
2*sqrt(abs(a))*B*a*cosh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)^2-a
^2*sqrt(abs(a))*B*a*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)^3+sqrt(3)*abs(a)*a^2*sqrt(abs(a
))*B*cos(re(acos(a/2/abs(a)))/2)^3*cosh(im(acos(a/2/abs(a)))/2)^3-3*sqrt(3)*abs(a)*a^2*sqrt(abs(a))*B*cos(re(a
cos(a/2/abs(a)))/2)^3*cosh(im(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)+3*sqrt(3)*abs(a)*a^2*sqrt(ab
s(a))*B*cos(re(acos(a/2/abs(a)))/2)^3*cosh(im(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)^2-sqrt(3)*abs(
a)*a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)^3-3*sqrt(3)*abs(a)*a^2*sqrt(a
bs(a))*B*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)^3*sin(re(acos(a/2/abs(a)))/2)^2+9*sqrt(3)*ab
s(a)*a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)
^2*sinh(im(acos(a/2/abs(a)))/2)-9*sqrt(3)*abs(a)*a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a
/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)^2+3*sqrt(3)*abs(a)*a^2*sqrt(abs(a))*
B*cos(re(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)^3)*1/4/sqrt(3)/a^4*ln
(x^2+2*sqrt(abs(a))*cos(acos(a*1/2/abs(a))/2)*x+sqrt(abs(a))*sqrt(abs(a)))-(-abs(a)*sqrt(abs(a))*A*a*cos(re(ac
os(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)+abs(a)*sqrt(abs(a))*A*a*cos(re(acos(a/2/abs(a)))/2)*sinh(im(ac
os(a/2/abs(a)))/2)-sqrt(3)*a^2*sqrt(abs(a))*A*cosh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)+sqrt(3)
*a^2*sqrt(abs(a))*A*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)-a^2*sqrt(abs(a))*B*a*cos(re(acos(
a/2/abs(a)))/2)^3*cosh(im(acos(a/2/abs(a)))/2)^3+3*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)^3*cosh(im(
acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)-3*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)^3*cosh(
im(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)^2+a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)^3*sinh
(im(acos(a/2/abs(a)))/2)^3+3*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)^3*s
in(re(acos(a/2/abs(a)))/2)^2-9*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)*cosh(im(acos(a/2/abs(a)))/2)^2
*sin(re(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)+9*a^2*sqrt(abs(a))*B*a*cos(re(acos(a/2/abs(a)))/2)
*cosh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)^2-3*a^2*sqrt(abs(a))*
B*a*cos(re(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^2*sinh(im(acos(a/2/abs(a)))/2)^3-3*sqrt(3)*abs(a)*
a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)^2*cosh(im(acos(a/2/abs(a)))/2)^3*sin(re(acos(a/2/abs(a)))/2)+9*
sqrt(3)*abs(a)*a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)^2*cosh(im(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2
/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)-9*sqrt(3)*abs(a)*a^2*sqrt(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)^2*co
sh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)^2+3*sqrt(3)*abs(a)*a^2*sqr
t(abs(a))*B*cos(re(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)*sinh(im(acos(a/2/abs(a)))/2)^3+sqrt(3)*a
bs(a)*a^2*sqrt(abs(a))*B*cosh(im(acos(a/2/abs(a)))/2)^3*sin(re(acos(a/2/abs(a)))/2)^3-3*sqrt(3)*abs(a)*a^2*sqr
t(abs(a))*B*cosh(im(acos(a/2/abs(a)))/2)^2*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)+3*sqrt(3
)*abs(a)*a^2*sqrt(abs(a))*B*cosh(im(acos(a/2/abs(a)))/2)*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)
))/2)^2-sqrt(3)*abs(a)*a^2*sqrt(abs(a))*B*sin(re(acos(a/2/abs(a)))/2)^3*sinh(im(acos(a/2/abs(a)))/2)^3)*1/2/sq
rt(3)/a^4*atan((x+cos(acos(a*1/2/abs(a))/2)*sqrt(abs(a)))/sin(acos(a*1/2/abs(a))/2)/sqrt(abs(a)))
________________________________________________________________________________________
maple [A] time = 0.03, size = 190, normalized size = 1.40 \begin {gather*} \frac {B \arctan \left (\frac {2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {B \arctan \left (\frac {-2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {3}\, B \ln \left (x^{2}+\sqrt {3}\, \sqrt {a}\, x +a \right )}{12 \sqrt {a}}+\frac {\sqrt {3}\, B \ln \left (-x^{2}+\sqrt {3}\, \sqrt {a}\, x -a \right )}{12 \sqrt {a}}+\frac {A \arctan \left (\frac {2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}-\frac {A \arctan \left (\frac {-2 x +\sqrt {3}\, \sqrt {a}}{\sqrt {a}}\right )}{2 a^{\frac {3}{2}}}+\frac {\sqrt {3}\, A \ln \left (x^{2}+\sqrt {3}\, \sqrt {a}\, x +a \right )}{12 a^{\frac {3}{2}}}-\frac {\sqrt {3}\, A \ln \left (-x^{2}+\sqrt {3}\, \sqrt {a}\, x -a \right )}{12 a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x^2+A)/(x^4-a*x^2+a^2),x)
[Out]
1/12/a^(1/2)*ln(-x^2+3^(1/2)*a^(1/2)*x-a)*B*3^(1/2)-1/12/a^(3/2)*ln(-x^2+3^(1/2)*a^(1/2)*x-a)*A*3^(1/2)-1/2/a^
(1/2)*arctan((-2*x+3^(1/2)*a^(1/2))/a^(1/2))*B-1/2/a^(3/2)*arctan((-2*x+3^(1/2)*a^(1/2))/a^(1/2))*A-1/12/a^(1/
2)*ln(x^2+3^(1/2)*a^(1/2)*x+a)*B*3^(1/2)+1/12/a^(3/2)*ln(x^2+3^(1/2)*a^(1/2)*x+a)*A*3^(1/2)+1/2/a^(1/2)*arctan
((2*x+3^(1/2)*a^(1/2))/a^(1/2))*B+1/2/a^(3/2)*arctan((2*x+3^(1/2)*a^(1/2))/a^(1/2))*A
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x^{2} + A}{x^{4} - a x^{2} + a^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
mupad [B] time = 4.59, size = 1007, normalized size = 7.40 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
________________________________________________________________________________________
sympy [A] time = 1.91, size = 172, normalized size = 1.26 \begin {gather*} \operatorname {RootSum} {\left (144 t^{4} a^{6} + t^{2} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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))))
________________________________________________________________________________________