3.39 \(\int \frac{1+2 x^2}{1+b x^2+4 x^4} \, dx\)

Optimal. Leaf size=62 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4-b}+4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}}-\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}-4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}} \]

[Out]

-(ArcTan[(Sqrt[4 - b] - 4*x)/Sqrt[4 + b]]/Sqrt[4 + b]) + ArcTan[(Sqrt[4 - b] + 4*x)/Sqrt[4 + b]]/Sqrt[4 + b]

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Rubi [A]  time = 0.0579177, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4-b}+4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}}-\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}-4 x}{\sqrt{b+4}}\right )}{\sqrt{b+4}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(1 + b*x^2 + 4*x^4),x]

[Out]

-(ArcTan[(Sqrt[4 - b] - 4*x)/Sqrt[4 + b]]/Sqrt[4 + b]) + ArcTan[(Sqrt[4 - b] + 4*x)/Sqrt[4 + b]]/Sqrt[4 + b]

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[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, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 618

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

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])

Rubi steps

\begin{align*} \int \frac{1+2 x^2}{1+b x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\frac{1}{2} \sqrt{4-b} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{1}{2} \sqrt{4-b} x+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{4} (-4-b)-x^2} \, dx,x,-\frac{\sqrt{4-b}}{2}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{4} (-4-b)-x^2} \, dx,x,\frac{\sqrt{4-b}}{2}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}-4 x}{\sqrt{4+b}}\right )}{\sqrt{4+b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{4-b}+4 x}{\sqrt{4+b}}\right )}{\sqrt{4+b}}\\ \end{align*}

Mathematica [B]  time = 0.0589532, size = 126, normalized size = 2.03 \[ \frac{\frac{\left (\sqrt{b^2-16}-b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right )}{\sqrt{b-\sqrt{b^2-16}}}+\frac{\left (\sqrt{b^2-16}+b-4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right )}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2)/(1 + b*x^2 + 4*x^4),x]

[Out]

(((4 - b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b - Sqrt[-16 + b^2]]])/Sqrt[b - Sqrt[-16 + b^2]] + ((-4
+ b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b + Sqrt[-16 + b^2]]])/Sqrt[b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sq
rt[-16 + b^2])

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Maple [B]  time = 0.137, size = 277, normalized size = 4.5 \begin{align*} -4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }+{\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+{b\arctan \left ( 4\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}+4\,{\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ) }+{\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}}-{b\arctan \left ( 4\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-4 \right ) \left ( 4+b \right ) }+2\,b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/((b-4)*(4+b))^(1/2)/(2*((b-4)*(4+b))^(1/2)+2*b)^(1/2)*arctan(4*x/(2*((b-4)*(4+b))^(1/2)+2*b)^(1/2))+1/(2*((
b-4)*(4+b))^(1/2)+2*b)^(1/2)*arctan(4*x/(2*((b-4)*(4+b))^(1/2)+2*b)^(1/2))+1/((b-4)*(4+b))^(1/2)/(2*((b-4)*(4+
b))^(1/2)+2*b)^(1/2)*arctan(4*x/(2*((b-4)*(4+b))^(1/2)+2*b)^(1/2))*b+4/((b-4)*(4+b))^(1/2)/(-2*((b-4)*(4+b))^(
1/2)+2*b)^(1/2)*arctan(4*x/(-2*((b-4)*(4+b))^(1/2)+2*b)^(1/2))+1/(-2*((b-4)*(4+b))^(1/2)+2*b)^(1/2)*arctan(4*x
/(-2*((b-4)*(4+b))^(1/2)+2*b)^(1/2))-1/((b-4)*(4+b))^(1/2)/(-2*((b-4)*(4+b))^(1/2)+2*b)^(1/2)*arctan(4*x/(-2*(
(b-4)*(4+b))^(1/2)+2*b)^(1/2))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + b x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.35349, size = 286, normalized size = 4.61 \begin{align*} \left [-\frac{\sqrt{-b - 4} \log \left (\frac{4 \, x^{4} -{\left (b + 8\right )} x^{2} - 2 \,{\left (2 \, x^{3} - x\right )} \sqrt{-b - 4} + 1}{4 \, x^{4} + b x^{2} + 1}\right )}{2 \,{\left (b + 4\right )}}, \frac{\sqrt{b + 4} \arctan \left (\frac{4 \, x^{3} +{\left (b + 2\right )} x}{\sqrt{b + 4}}\right ) + \sqrt{b + 4} \arctan \left (\frac{2 \, x}{\sqrt{b + 4}}\right )}{b + 4}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-b - 4)*log((4*x^4 - (b + 8)*x^2 - 2*(2*x^3 - x)*sqrt(-b - 4) + 1)/(4*x^4 + b*x^2 + 1))/(b + 4), (s
qrt(b + 4)*arctan((4*x^3 + (b + 2)*x)/sqrt(b + 4)) + sqrt(b + 4)*arctan(2*x/sqrt(b + 4)))/(b + 4)]

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Sympy [A]  time = 0.27038, size = 95, normalized size = 1.53 \begin{align*} - \frac{\sqrt{- \frac{1}{b + 4}} \log{\left (x^{2} + x \left (- \frac{b \sqrt{- \frac{1}{b + 4}}}{2} - 2 \sqrt{- \frac{1}{b + 4}}\right ) - \frac{1}{2} \right )}}{2} + \frac{\sqrt{- \frac{1}{b + 4}} \log{\left (x^{2} + x \left (\frac{b \sqrt{- \frac{1}{b + 4}}}{2} + 2 \sqrt{- \frac{1}{b + 4}}\right ) - \frac{1}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(b + 4))*log(x**2 + x*(-b*sqrt(-1/(b + 4))/2 - 2*sqrt(-1/(b + 4))) - 1/2)/2 + sqrt(-1/(b + 4))*log(x*
*2 + x*(b*sqrt(-1/(b + 4))/2 + 2*sqrt(-1/(b + 4))) - 1/2)/2

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Giac [C]  time = 1.44781, size = 3352, normalized size = 54.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+b*x^2+1),x, algorithm="giac")

[Out]

1/4*(3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*cosh
(1/2*imag_part(arcsin(1/4*b)))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2
- 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arcsin(1/4*b)))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3 - 9*(s
qrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*cosh(1/2*imag
_part(arcsin(1/4*b)))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))) + 3*(sqr
t(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arcsin(1/4*b)))^2*sin(5/4*pi + 1/2*real_p
art(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*b))) + 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt
(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(5/4*pi + 1/2*real_par
t(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b)))^2 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2
))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1
/4*b)))^2 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))
^2*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b)))^3 + (sqrt(2)*b^2 + sqrt(2)*sq
rt(b^2 - 16)*b - 16*sqrt(2))*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*b)))^3
+ (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(5/4*pi + 1/2*r
eal_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(5/4*pi + 1/2*real_part(ar
csin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))))*arctan(-4*(1/4)^(3/4)*((1/4)^(1/4)*cos(5/4*pi + 1/2*arcsin(1
/4*b)) - x)/sin(5/4*pi + 1/2*arcsin(1/4*b)))/(b^2 - 16) + 1/4*(3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*
sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*cosh(1/2*imag_part(arcsin(1/4*b)))^3*sin(1/4*pi + 1/2*re
al_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arcsin(1/4*
b)))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^3 - 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*
cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*cosh(1/2*imag_part(arcsin(1/4*b)))^2*sin(1/4*pi + 1/2*real_part(a
rcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))) + 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*co
sh(1/2*imag_part(arcsin(1/4*b)))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*
b))) + 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*co
sh(1/2*imag_part(arcsin(1/4*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b)))
^2 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(1/4*pi + 1
/2*real_part(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*b)))^2 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*
b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1
/2*imag_part(arcsin(1/4*b)))^3 + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(1/4*pi + 1/2*real_p
art(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*b)))^3 + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt
(2))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sq
rt(b^2 - 16)*b - 16*sqrt(2))*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))))*ar
ctan(-4*(1/4)^(3/4)*((1/4)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/4*b)) - x)/sin(1/4*pi + 1/2*arcsin(1/4*b)))/(b^2 -
16) - 1/8*((sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*
cosh(1/2*imag_part(arcsin(1/4*b)))^3 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/
2*real_part(arcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2
- 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*cosh(1/
2*imag_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(arcsin(1/4*b))) + 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b -
16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))^2*sin(5/4*pi + 1/2*
real_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(arcsin(1/4*b))) + 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 1
6*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*cosh(1/2*imag_part(arcsin(1/4*b)))*sinh(1/2*imag_part(
arcsin(1/4*b)))^2 - 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(
1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(ar
csin(1/4*b)))^2 - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*
b)))^3*sinh(1/2*imag_part(arcsin(1/4*b)))^3 + 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*
pi + 1/2*real_part(arcsin(1/4*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(arcsin(1/4*
b)))^3 + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/4*b)))*cosh
(1/2*imag_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(5/4*pi + 1/2*real_p
art(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))))*log(-2*(1/4)^(1/4)*x*cos(5/4*pi + 1/2*arcsin(1/4*b)) +
x^2 + 1/2)/(b^2 - 16) - 1/8*((sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part
(arcsin(1/4*b)))^3*cosh(1/2*imag_part(arcsin(1/4*b)))^3 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(
2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))^3*sin(1/4*pi + 1/2*real_part
(arcsin(1/4*b)))^2 - 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin
(1/4*b)))^3*cosh(1/2*imag_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(arcsin(1/4*b))) + 9*(sqrt(2)*b^2 + sqrt(2)
*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))^
2*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*sinh(1/2*imag_part(arcsin(1/4*b))) + 3*(sqrt(2)*b^2 + sqrt(2)*s
qrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^3*cosh(1/2*imag_part(arcsin(1/4*b)))*
sinh(1/2*imag_part(arcsin(1/4*b)))^2 - 9*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/
2*real_part(arcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*si
nh(1/2*imag_part(arcsin(1/4*b)))^2 - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*re
al_part(arcsin(1/4*b)))^3*sinh(1/2*imag_part(arcsin(1/4*b)))^3 + 3*(sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 1
6*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/4*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/4*b)))^2*sinh(1/2*im
ag_part(arcsin(1/4*b)))^3 + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/4*pi + 1/2*real_part(a
rcsin(1/4*b)))*cosh(1/2*imag_part(arcsin(1/4*b))) - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(
1/4*pi + 1/2*real_part(arcsin(1/4*b)))*sinh(1/2*imag_part(arcsin(1/4*b))))*log(-2*(1/4)^(1/4)*x*cos(1/4*pi + 1
/2*arcsin(1/4*b)) + x^2 + 1/2)/(b^2 - 16)