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

Optimal. Leaf size=66 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b+4}+4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b+4}-4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}} \]

[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 = 0.0579648, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1161, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b+4}+4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b+4}-4 x}{\sqrt{4-b}}\right )}{\sqrt{4-b}} \]

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 = 0.0583946, size = 134, normalized size = 2.03 \[ \frac{\frac{\left (\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{-\sqrt{b^2-16}-b}}\right )}{\sqrt{-\sqrt{b^2-16}-b}}+\frac{\left (\sqrt{b^2-16}-b-4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}-b}}\right )}{\sqrt{\sqrt{b^2-16}-b}}}{\sqrt{2} \sqrt{b^2-16}} \]

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
]*Sqrt[-16 + b^2])

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Maple [B]  time = 0.106, size = 277, normalized size = 4.2 \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 = 1.26934, size = 298, normalized size = 4.52 \begin{align*} \left [\frac{\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 \, \sqrt{b - 4}}, \frac{\sqrt{-b + 4} \arctan \left (\frac{{\left (4 \, x^{3} -{\left (b - 2\right )} x\right )} \sqrt{-b + 4}}{b - 4}\right ) + \sqrt{-b + 4} \arctan \left (\frac{2 \, \sqrt{-b + 4} x}{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*log((4*x^4 + (b - 8)*x^2 - 2*(2*x^3 - x)*sqrt(b - 4) + 1)/(4*x^4 - b*x^2 + 1))/sqrt(b - 4), (sqrt(-b + 4)
*arctan((4*x^3 - (b - 2)*x)*sqrt(-b + 4)/(b - 4)) + sqrt(-b + 4)*arctan(2*sqrt(-b + 4)*x/(b - 4)))/(b - 4)]

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Sympy [A]  time = 0.28598, size = 83, normalized size = 1.26 \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 = 1.2722, size = 3069, normalized size = 46.5 \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(1/2*real_part(arccos(1/4*b)))^2*cosh(1/2*imag
_part(arccos(1/4*b)))^3*sin(1/2*real_part(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(
2))*cosh(1/2*imag_part(arccos(1/4*b)))^3*sin(1/2*real_part(arccos(1/4*b)))^3 - 9*(sqrt(2)*b^2 - sqrt(2)*sqrt(b
^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^2*cosh(1/2*imag_part(arccos(1/4*b)))^2*sin(1/2*real
_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b))) + 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt
(2))*cosh(1/2*imag_part(arccos(1/4*b)))^2*sin(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*imag_part(arccos(1/4*b)
)) + 9*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^2*cosh(1/2*imag
_part(arccos(1/4*b)))*sin(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b)))^2 - 3*(sqrt(2)*b^2
- sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/4*b)))*sin(1/2*real_part(arccos(1/4*b)))^
3*sinh(1/2*imag_part(arccos(1/4*b)))^2 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_
part(arccos(1/4*b)))^2*sin(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b)))^3 + (sqrt(2)*b^2 -
sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*imag_part(arccos(1/4*b)))
^3 + (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/4*b)))*sin(1/2*real_par
t(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(1/2*real_part(arccos(1/4*b)))*si
nh(1/2*imag_part(arccos(1/4*b))))*arctan(4*(1/4)^(3/4)*((1/4)^(1/4)*cos(1/2*arccos(1/4*b)) + x)/sin(1/2*arccos
(1/4*b)))/(b^2 - 16) + 1/4*(3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1
/4*b)))^2*cosh(1/2*imag_part(arccos(1/4*b)))^3*sin(1/2*real_part(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt
(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/4*b)))^3*sin(1/2*real_part(arccos(1/4*b)))^3 - 9*(sqrt(
2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^2*cosh(1/2*imag_part(arccos(
1/4*b)))^2*sin(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b))) + 3*(sqrt(2)*b^2 - sqrt(2)*sqr
t(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/4*b)))^2*sin(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*
imag_part(arccos(1/4*b))) + 9*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1
/4*b)))^2*cosh(1/2*imag_part(arccos(1/4*b)))*sin(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b
)))^2 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/4*b)))*sin(1/2*rea
l_part(arccos(1/4*b)))^3*sinh(1/2*imag_part(arccos(1/4*b)))^2 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16
*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^2*sin(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*
b)))^3 + (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*im
ag_part(arccos(1/4*b)))^3 + (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cosh(1/2*imag_part(arccos(1/
4*b)))*sin(1/2*real_part(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*sin(1/2*real_
part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b))))*arctan(-4*(1/4)^(3/4)*((1/4)^(1/4)*cos(1/2*arccos(1/4
*b)) - x)/sin(1/2*arccos(1/4*b)))/(b^2 - 16) + 1/8*((sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(
1/2*real_part(arccos(1/4*b)))^3*cosh(1/2*imag_part(arccos(1/4*b)))^3 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)
*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b)))^3*sin(1/2*real_part(arcc
os(1/4*b)))^2 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^3*co
sh(1/2*imag_part(arccos(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b))) + 9*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16
)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b)))^2*sin(1/2*real_part(arc
cos(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b))) + 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*co
s(1/2*real_part(arccos(1/4*b)))^3*cosh(1/2*imag_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b)))^2 - 9*
(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arc
cos(1/4*b)))*sin(1/2*real_part(arccos(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b)))^2 - (sqrt(2)*b^2 - sqrt(2)
*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*imag_part(arccos(1/4*b)))^3 + 3*(
sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*sin(1/2*real_part(arcco
s(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b)))^3 + (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(
1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 1
6*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b))))*log(2*(1/4)^(1/4)*x*cos(1/2*a
rccos(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(1/2*rea
l_part(arccos(1/4*b)))^3*cosh(1/2*imag_part(arccos(1/4*b)))^3 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16
*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b)))^3*sin(1/2*real_part(arccos(1/4*
b)))^2 - 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^3*cosh(1/2*
imag_part(arccos(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b))) + 9*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 1
6*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b)))^2*sin(1/2*real_part(arccos(1/4
*b)))^2*sinh(1/2*imag_part(arccos(1/4*b))) + 3*(sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*r
eal_part(arccos(1/4*b)))^3*cosh(1/2*imag_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b)))^2 - 9*(sqrt(2
)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4
*b)))*sin(1/2*real_part(arccos(1/4*b)))^2*sinh(1/2*imag_part(arccos(1/4*b)))^2 - (sqrt(2)*b^2 - sqrt(2)*sqrt(b
^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))^3*sinh(1/2*imag_part(arccos(1/4*b)))^3 + 3*(sqrt(2)
*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*real_part(arccos(1/4*b)))*sin(1/2*real_part(arccos(1/4*b
)))^2*sinh(1/2*imag_part(arccos(1/4*b)))^3 + (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(2))*cos(1/2*rea
l_part(arccos(1/4*b)))*cosh(1/2*imag_part(arccos(1/4*b))) - (sqrt(2)*b^2 - sqrt(2)*sqrt(b^2 - 16)*b - 16*sqrt(
2))*cos(1/2*real_part(arccos(1/4*b)))*sinh(1/2*imag_part(arccos(1/4*b))))*log(-2*(1/4)^(1/4)*x*cos(1/2*arccos(
1/4*b)) + x^2 + 1/2)/(b^2 - 16)