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3.2.49 \(\int \frac {1}{(d+e x^2)^2 (a+c x^4)^2} \, dx\) [149]

Optimal. Leaf size=864 \[ \frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2} \]

[Out]

1/2*e^4*x/d/(a*e^2+c*d^2)^2/(e*x^2+d)+1/4*c*x*(-2*c*d*e*x^2-a*e^2+c*d^2)/a/(a*e^2+c*d^2)^2/(c*x^4+a)+1/2*e^(7/
2)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(a*e^2+c*d^2)^2+1/4*c^(3/4)*e^2*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c
*d^2-a*e^2-4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/4*c^(3/4)*e^2*arctan(1+c^(1/4)*x*2^(1/2)/a
^(1/4))*(3*c*d^2-a*e^2-4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/16*c^(3/4)*arctan(-1+c^(1/4)*x
*2^(1/2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)+1/16*c^(3/4)*arctan(
1+c^(1/4)*x*2^(1/2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/32*c^(3
/4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-3*a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/(a*e^2+
c*d^2)^2*2^(1/2)+1/32*c^(3/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-3*a*e^2+2*d*e*a^(1/2)
*c^(1/2))/a^(7/4)/(a*e^2+c*d^2)^2*2^(1/2)-1/8*c^(3/4)*e^2*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(
3*c*d^2-a*e^2+4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+1/8*c^(3/4)*e^2*ln(a^(1/4)*c^(1/4)*x*2^(1
/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2-a*e^2+4*d*e*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^2+c*d^2)^3*2^(1/2)+4*c*e^(7/2)*arc
tan(x*e^(1/2)/d^(1/2))*d^(1/2)/(a*e^2+c*d^2)^3

________________________________________________________________________________________

Rubi [A]
time = 0.59, antiderivative size = 864, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1253, 205, 211, 1193, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac {4 c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2-4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+4 \sqrt {a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^2}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2))/(4*a*(c*d^2 + a*e^2)^2*(a +
c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(2*d^(3/2)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqr
t[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e -
3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^
2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)
^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]
*a^(7/4)*(c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d
*e - 3*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) +
 (c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + S
qrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
 + 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
 x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
 Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 1193

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)*((a + c*x^4)^(p + 1)/
(4*a*(p + 1))), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x],
 x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1253

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a +
 c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, p, q}, x] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0])

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac {e^4}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}+\frac {4 c d e^4}{\left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}+\frac {c \left (c d^2-a e^2-2 c d e x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^2}-\frac {c e^2 \left (-3 c d^2+a e^2+4 c d e x^2\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {\left (c e^2\right ) \int \frac {-3 c d^2+a e^2+4 c d e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (4 c d e^4\right ) \int \frac {1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {c \int \frac {c d^2-a e^2-2 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {e^4 \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^3}-\frac {c \int \frac {-3 \left (c d^2-a e^2\right )+2 c d e x^2}{a+c x^4} \, dx}{4 a \left (c d^2+a e^2\right )^2}+\frac {e^4 \int \frac {1}{d+e x^2} \, dx}{2 d \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \sqrt {a} \left (c d^2+a e^2\right )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {\left (c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {\left (c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {\left (c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ &=\frac {e^4 x}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {c x \left (c d^2-a e^2-2 c d e x^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )}+\frac {4 c \sqrt {d} e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2-4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} e^2 \left (3 c d^2+4 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac {c^{3/4} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^2+a e^2\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 540, normalized size = 0.62 \begin {gather*} \frac {\frac {16 e^4 \left (c d^2+a e^2\right ) x}{d \left (d+e x^2\right )}+\frac {8 c \left (c d^2+a e^2\right ) x \left (-a e^2+c d \left (d-2 e x^2\right )\right )}{a \left (a+c x^4\right )}+\frac {16 e^{7/2} \left (9 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 \sqrt {2} c^{3/4} \left (-3 c^2 d^4+2 \sqrt {a} c^{3/2} d^3 e-12 a c d^2 e^2+18 a^{3/2} \sqrt {c} d e^3+7 a^2 e^4\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {2 \sqrt {2} c^{3/4} \left (-3 c^2 d^4+2 \sqrt {a} c^{3/2} d^3 e-12 a c d^2 e^2+18 a^{3/2} \sqrt {c} d e^3+7 a^2 e^4\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {\sqrt {2} c^{3/4} \left (3 c^2 d^4+2 \sqrt {a} c^{3/2} d^3 e+12 a c d^2 e^2+18 a^{3/2} \sqrt {c} d e^3-7 a^2 e^4\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} c^{3/4} \left (3 c^2 d^4+2 \sqrt {a} c^{3/2} d^3 e+12 a c d^2 e^2+18 a^{3/2} \sqrt {c} d e^3-7 a^2 e^4\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}}{32 \left (c d^2+a e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((16*e^4*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (8*c*(c*d^2 + a*e^2)*x*(-(a*e^2) + c*d*(d - 2*e*x^2)))/(a*(a + c
*x^4)) + (16*e^(7/2)*(9*c*d^2 + a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 +
 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*
x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*
Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*
Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c
^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18
*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^
2 + a*e^2)^3)

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Maple [A]
time = 0.25, size = 402, normalized size = 0.47

method result size
default \(\frac {e^{4} \left (\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a \,e^{2}+9 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}-\frac {c \left (\frac {\frac {c d e \left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{2 a}+\frac {\left (a^{2} e^{4}-c^{2} d^{4}\right ) x}{4 a}}{c \,x^{4}+a}+\frac {\frac {\left (7 a^{2} e^{4}-12 a c \,d^{2} e^{2}-3 c^{2} d^{4}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (18 a c d \,e^{3}+2 c^{2} d^{3} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{3}}\) \(402\)
risch \(\text {Expression too large to display}\) \(3814\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^4/(a*e^2+c*d^2)^3*(1/2*(a*e^2+c*d^2)/d*x/(e*x^2+d)+1/2*(a*e^2+9*c*d^2)/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))
)-c/(a*e^2+c*d^2)^3*((1/2*c*d*e*(a*e^2+c*d^2)/a*x^3+1/4*(a^2*e^4-c^2*d^4)/a*x)/(c*x^4+a)+1/4/a*(1/8*(7*a^2*e^4
-12*a*c*d^2*e^2-3*c^2*d^4)*(a/c)^(1/4)/a*2^(1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*
x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/8*(18*a*c*d*e^3
+2*c^2*d^3*e)/c/(a/c)^(1/4)*2^(1/2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/
c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))))

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Maxima [A]
time = 0.54, size = 705, normalized size = 0.82 \begin {gather*} \frac {{\left (9 \, c d^{2} e^{4} + a e^{6}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, {\left (c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6}\right )} \sqrt {d}} + \frac {c {\left (\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} - 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} - 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} - 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} - 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} + 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} + 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {5}{2}} d^{4} + 2 \, \sqrt {a} c^{2} d^{3} e + 12 \, a c^{\frac {3}{2}} d^{2} e^{2} + 18 \, a^{\frac {3}{2}} c d e^{3} - 7 \, a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )}} - \frac {2 \, {\left (c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{5} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} - {\left (c^{2} d^{4} - a c d^{2} e^{2} + 2 \, a^{2} e^{4}\right )} x}{4 \, {\left (a^{2} c^{2} d^{6} + 2 \, a^{3} c d^{4} e^{2} + {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{6} + a^{4} d^{2} e^{4} + {\left (a c^{3} d^{6} + 2 \, a^{2} c^{2} d^{4} e^{2} + a^{3} c d^{2} e^{4}\right )} x^{4} + {\left (a^{2} c^{2} d^{5} e + 2 \, a^{3} c d^{3} e^{3} + a^{4} d e^{5}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(9*c*d^2*e^4 + a*e^6)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/((c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a
^3*d*e^6)*sqrt(d)) + 1/32*c*(2*sqrt(2)*(3*c^(5/2)*d^4 - 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 - 18*a^(3/2
)*c*d*e^3 - 7*a^2*sqrt(c)*e^4)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)
))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(3*c^(5/2)*d^4 - 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2
*e^2 - 18*a^(3/2)*c*d*e^3 - 7*a^2*sqrt(c)*e^4)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt
(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(3*c^(5/2)*d^4 + 2*sqrt(a)*c^2*d^3*e + 12
*a*c^(3/2)*d^2*e^2 + 18*a^(3/2)*c*d*e^3 - 7*a^2*sqrt(c)*e^4)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqr
t(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(3*c^(5/2)*d^4 + 2*sqrt(a)*c^2*d^3*e + 12*a*c^(3/2)*d^2*e^2 + 18*a^(3/2)*c*d
*e^3 - 7*a^2*sqrt(c)*e^4)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c^3*d^6
 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6) - 1/4*(2*(c^2*d^2*e^2 - a*c*e^4)*x^5 + (c^2*d^3*e + a*c*d*e^
3)*x^3 - (c^2*d^4 - a*c*d^2*e^2 + 2*a^2*e^4)*x)/(a^2*c^2*d^6 + 2*a^3*c*d^4*e^2 + (a*c^3*d^5*e + 2*a^2*c^2*d^3*
e^3 + a^3*c*d*e^5)*x^6 + a^4*d^2*e^4 + (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 + a^3*c*d^2*e^4)*x^4 + (a^2*c^2*d^5*e +
2*a^3*c*d^3*e^3 + a^4*d*e^5)*x^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7251 vs. \(2 (660) = 1320\).
time = 97.76, size = 14534, normalized size = 16.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(8*c^3*d^4*x^5*e^2 + 4*c^3*d^5*x^3*e - 4*c^3*d^6*x + 8*a*c^2*d^3*x^3*e^3 + 4*a^2*c*d*x^3*e^5 - 4*a^2*c*
d^2*x*e^4 - (a*c^4*d^8*x^4 + a^2*c^3*d^8 + (a^4*c*d*x^6 + a^5*d*x^2)*e^7 + (a^4*c*d^2*x^4 + a^5*d^2)*e^6 + 3*(
a^3*c^2*d^3*x^6 + a^4*c*d^3*x^2)*e^5 + 3*(a^3*c^2*d^4*x^4 + a^4*c*d^4)*e^4 + 3*(a^2*c^3*d^5*x^6 + a^3*c^2*d^5*
x^2)*e^3 + 3*(a^2*c^3*d^6*x^4 + a^3*c^2*d^6)*e^2 + (a*c^4*d^7*x^6 + a^2*c^3*d^7*x^2)*e)*sqrt((12*c^5*d^7*e + 1
56*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d
^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c
^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10
+ 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2
+ 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6
*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 1
2*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6
+ 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*log(81*c^7*d^10*x + 1053*a*c^6*d^8*x*e^2 + 3602*a^2*c^5*d
^6*x*e^4 - 2958*a^3*c^4*d^4*x*e^6 - 23667*a^4*c^3*d^2*x*e^8 + 2401*a^5*c^2*x*e^10 + (27*a^2*c^7*d^12 + 312*a^3
*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 34
3*a^8*c*e^12 + 2*(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*
c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^
14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 2450
04*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a
^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*
e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18
*c*d^2*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 +
(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^
2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6
- 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8
*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8
*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 +
 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^
10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) + (a*c^
4*d^8*x^4 + a^2*c^3*d^8 + (a^4*c*d*x^6 + a^5*d*x^2)*e^7 + (a^4*c*d^2*x^4 + a^5*d^2)*e^6 + 3*(a^3*c^2*d^3*x^6 +
 a^4*c*d^3*x^2)*e^5 + 3*(a^3*c^2*d^4*x^4 + a^4*c*d^4)*e^4 + 3*(a^2*c^3*d^5*x^6 + a^3*c^2*d^5*x^2)*e^3 + 3*(a^2
*c^3*d^6*x^4 + a^3*c^2*d^6)*e^2 + (a*c^4*d^7*x^6 + a^2*c^3*d^7*x^2)*e)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3
+ 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c
^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 51
64*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*
d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^2
0*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*
a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22
 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*
e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*log(81*c^7*d^10*x + 1053*a*c^6*d^8*x*e^2 + 3602*a^2*c^5*d^6*x*e^4 - 2958*a
^3*c^4*d^4*x*e^6 - 23667*a^4*c^3*d^2*x*e^8 + 2401*a^5*c^2*x*e^10 - (27*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 8
43*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 + 2*
(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e^9 + 141
*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2
*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^
12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4
+ 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 3.30, size = 855, normalized size = 0.99 \begin {gather*} \frac {{\left (9 \, c d^{2} e^{4} + a e^{6}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, {\left (c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6}\right )} \sqrt {d}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 18 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{6} + 3 \, \sqrt {2} a^{3} c^{3} d^{4} e^{2} + 3 \, \sqrt {2} a^{4} c^{2} d^{2} e^{4} + \sqrt {2} a^{5} c e^{6}\right )}} + \frac {{\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 18 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{6} + 3 \, \sqrt {2} a^{3} c^{3} d^{4} e^{2} + 3 \, \sqrt {2} a^{4} c^{2} d^{2} e^{4} + \sqrt {2} a^{5} c e^{6}\right )}} + \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 18 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{6} + 3 \, a^{3} c^{3} d^{4} e^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )}} - \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} + 12 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 18 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{6} + 3 \, a^{3} c^{3} d^{4} e^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} + a^{5} c e^{6}\right )}} - \frac {2 \, c^{2} d^{2} x^{5} e^{2} + c^{2} d^{3} x^{3} e - 2 \, a c x^{5} e^{4} - c^{2} d^{4} x + a c d x^{3} e^{3} + a c d^{2} x e^{2} - 2 \, a^{2} x e^{4}}{4 \, {\left (a c^{2} d^{5} + 2 \, a^{2} c d^{3} e^{2} + a^{3} d e^{4}\right )} {\left (c x^{6} e + c d x^{4} + a x^{2} e + a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(9*c*d^2*e^4 + a*e^6)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/((c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a
^3*d*e^6)*sqrt(d)) + 1/8*(3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e -
 7*(a*c^3)^(1/4)*a^2*c*e^4 - 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1
/4))/(sqrt(2)*a^2*c^4*d^6 + 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/8*(
3*(a*c^3)^(1/4)*c^3*d^4 + 12*(a*c^3)^(1/4)*a*c^2*d^2*e^2 - 2*(a*c^3)^(3/4)*c*d^3*e - 7*(a*c^3)^(1/4)*a^2*c*e^4
 - 18*(a*c^3)^(3/4)*a*d*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^4*d^6
+ 3*sqrt(2)*a^3*c^3*d^4*e^2 + 3*sqrt(2)*a^4*c^2*d^2*e^4 + sqrt(2)*a^5*c*e^6) + 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c
^3*d^4 + 12*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + 2*sqrt(2)*(a*c^3)^(3/4)*c*d^3*e - 7*sqrt(2)*(a*c^3)^(1/4)*a^
2*c*e^4 + 18*sqrt(2)*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^6 + 3*a^3*
c^3*d^4*e^2 + 3*a^4*c^2*d^2*e^4 + a^5*c*e^6) - 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^3*d^4 + 12*sqrt(2)*(a*c^3)^(1/4
)*a*c^2*d^2*e^2 + 2*sqrt(2)*(a*c^3)^(3/4)*c*d^3*e - 7*sqrt(2)*(a*c^3)^(1/4)*a^2*c*e^4 + 18*sqrt(2)*(a*c^3)^(3/
4)*a*d*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^6 + 3*a^3*c^3*d^4*e^2 + 3*a^4*c^2*d^2*e^4
+ a^5*c*e^6) - 1/4*(2*c^2*d^2*x^5*e^2 + c^2*d^3*x^3*e - 2*a*c*x^5*e^4 - c^2*d^4*x + a*c*d*x^3*e^3 + a*c*d^2*x*
e^2 - 2*a^2*x*e^4)/((a*c^2*d^5 + 2*a^2*c*d^3*e^2 + a^3*d*e^4)*(c*x^6*e + c*d*x^4 + a*x^2*e + a*d))

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Mupad [B]
time = 8.33, size = 2500, normalized size = 2.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(2*a^2*e^4 + c^2*d^4 - a*c*d^2*e^2))/(4*a*d*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (c*e*x^3)/(4*a*(a*e^2 +
 c*d^2)) + (c*e^2*x^5*(a*e^2 - c*d^2))/(2*a*d*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a*d + a*e*x^2 + c*d*x^4 +
 c*e*x^6) + atan(((((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14
*e^7 + 296832*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e
^15 - 1254784*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^1
4 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10
 + 28*a^10*c^2*d^6*e^12)) - (((65536*a^15*c^4*d*e^24 - 24576*a^4*c^15*d^23*e^2 - 212992*a^5*c^14*d^21*e^4 - 35
2256*a^6*c^13*d^19*e^6 + 1966080*a^7*c^12*d^17*e^8 + 10960896*a^8*c^11*d^15*e^10 + 25460736*a^9*c^10*d^13*e^12
 + 34750464*a^10*c^9*d^11*e^14 + 30081024*a^11*c^8*d^9*e^16 + 16588800*a^12*c^7*d^7*e^18 + 5554176*a^13*c^6*d^
5*e^20 + 991232*a^14*c^5*d^3*e^22)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2
 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^1
2)) - (x*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 -
156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(
1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^1
0*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2)*(65536*a^6*c^15*d^24*e^3 + 589
824*a^7*c^14*d^22*e^5 + 2293760*a^8*c^13*d^20*e^7 + 4915200*a^9*c^12*d^18*e^9 + 5898240*a^10*c^11*d^16*e^11 +
2752512*a^11*c^10*d^14*e^13 - 2752512*a^12*c^9*d^12*e^15 - 5898240*a^13*c^8*d^10*e^17 - 4915200*a^14*c^7*d^8*e
^19 - 2293760*a^15*c^6*d^6*e^21 - 589824*a^16*c^5*d^4*e^23 - 65536*a^17*c^4*d^2*e^25))/(128*(a^4*c^8*d^18 + a^
12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*
d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)
^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(
-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 +
a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2
*d^4*e^8)))^(1/2) - (x*(4096*a^12*c^5*d*e^22 - 1152*a^2*c^15*d^21*e^2 - 15232*a^3*c^14*d^19*e^4 - 78336*a^4*c^
13*d^17*e^6 - 140800*a^5*c^12*d^15*e^8 + 489728*a^6*c^11*d^13*e^10 + 2219776*a^7*c^10*d^11*e^12 + 3155456*a^8*
c^9*d^9*e^14 + 1901056*a^9*c^8*d^7*e^16 + 362368*a^10*c^7*d^5*e^18 - 32640*a^11*c^6*d^3*e^20))/(128*(a^4*c^8*d
^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*
a^8*c^4*d^10*e^8 + 56*a^9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-
a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d
^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13
*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*
a^11*c^2*d^4*e^8)))^(1/2))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*c^5*d^7*e + 25
2*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2) - 492*a^3*c*d^
2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6*a^12*c*d^2*e^
10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2) - (x*(81*c^1
3*d^14*e^5 - 392*a^7*c^6*e^19 + 1206*a*c^12*d^12*e^7 + 12247*a^2*c^11*d^10*e^9 + 58636*a^3*c^10*d^8*e^11 + 114
927*a^4*c^9*d^6*e^13 - 1306*a^5*c^8*d^4*e^15 - 3575*a^6*c^7*d^2*e^17))/(128*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*
a^11*c*d^4*e^14 + 8*a^5*c^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*d^12*e^6 + 70*a^8*c^4*d^10*e^8 + 56*a^
9*c^3*d^8*e^10 + 28*a^10*c^2*d^6*e^12)))*(-(49*a^4*e^8*(-a^7*c^3)^(1/2) + 9*c^4*d^8*(-a^7*c^3)^(1/2) - 12*a^4*
c^5*d^7*e + 252*a^7*c^2*d*e^7 - 156*a^5*c^4*d^5*e^3 - 404*a^6*c^3*d^3*e^5 + 68*a*c^3*d^6*e^2*(-a^7*c^3)^(1/2)
- 492*a^3*c*d^2*e^6*(-a^7*c^3)^(1/2) + 30*a^2*c^2*d^4*e^4*(-a^7*c^3)^(1/2))/(256*(a^13*e^12 + a^7*c^6*d^12 + 6
*a^12*c*d^2*e^10 + 6*a^8*c^5*d^10*e^2 + 15*a^9*c^4*d^8*e^4 + 20*a^10*c^3*d^6*e^6 + 15*a^11*c^2*d^4*e^8)))^(1/2
)*1i - (((3584*a^10*c^5*e^21 + 1152*a*c^14*d^18*e^3 + 13184*a^2*c^13*d^16*e^5 + 54912*a^3*c^12*d^14*e^7 + 2968
32*a^4*c^11*d^12*e^9 + 1282432*a^5*c^10*d^10*e^11 + 769152*a^6*c^9*d^8*e^13 - 1421440*a^7*c^8*d^6*e^15 - 12547
84*a^8*c^7*d^4*e^17 - 89088*a^9*c^6*d^2*e^19)/(512*(a^4*c^8*d^18 + a^12*d^2*e^16 + 8*a^11*c*d^4*e^14 + 8*a^5*c
^7*d^16*e^2 + 28*a^6*c^6*d^14*e^4 + 56*a^7*c^5*...

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