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

Optimal. Leaf size=66 \[ \frac{\log \left (\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}}-\frac{\log \left (-\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}} \]

[Out]

-Log[1 - Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b]) + Log[1 + Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b])

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Rubi [A]  time = 0.028782, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1164, 628} \[ \frac{\log \left (\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}}-\frac{\log \left (-\sqrt{4-b} x+2 x^2+1\right )}{2 \sqrt{4-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 - Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b]) + Log[1 + Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - b])

Rule 1164

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*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, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[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]

Rubi steps

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

Mathematica [A]  time = 0.0699397, size = 127, normalized size = 1.92 \[ \frac{\frac{\left (-\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{b-\sqrt{b^2-16}}}\right )}{\sqrt{b-\sqrt{b^2-16}}}-\frac{\left (\sqrt{b^2-16}+b+4\right ) \tan ^{-1}\left (\frac{2 \sqrt{2} x}{\sqrt{\sqrt{b^2-16}+b}}\right )}{\sqrt{\sqrt{b^2-16}+b}}}{\sqrt{2} \sqrt{b^2-16}} \]

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]*Sqr
t[-16 + b^2])

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Maple [B]  time = 0.102, size = 279, 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 = 1.30515, size = 286, normalized size = 4.33 \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 = 0.26749, size = 94, normalized size = 1.42 \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 = 1.2965, size = 3352, normalized size = 50.79 \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*cos
h(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*(
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*ima
g_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*(sq
rt(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_
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*sqr
t(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_pa
rt(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)*s
qrt(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*
real_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(a
rcsin(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*r
eal_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(
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))*c
osh(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*c
osh(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_
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*sqr
t(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)*s
qrt(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))))*a
rctan(-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 -
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)))*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(a
rcsin(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)))*cos
h(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_
part(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_par
t(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_par
t(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(arcsi
n(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)*
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)))
*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*s
inh(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*r
eal_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(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*i
mag_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(
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
(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)

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