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

Optimal. Leaf size=160 \[ -\frac{\left (1+\sqrt{2}\right ) \log \left (-\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}+\frac{\left (1+\sqrt{2}\right ) \log \left (\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}+\frac{\left (1-\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}}-\frac{\left (1-\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}} \]

[Out]

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

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Rubi [A]  time = 0.119352, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1169, 634, 618, 204, 628} \[ -\frac{\left (1+\sqrt{2}\right ) \log \left (-\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}+\frac{\left (1+\sqrt{2}\right ) \log \left (\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}+\frac{\left (1-\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}}-\frac{\left (1-\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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])

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

Mathematica [A]  time = 0.0911318, size = 137, normalized size = 0.86 \[ \frac{\frac{\left (-\sqrt{b^2-4}+b+2 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right )}{\sqrt{b-\sqrt{b^2-4}}}-\frac{\left (\sqrt{b^2-4}+b+2 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right )}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.106, size = 285, normalized size = 1.8 \begin{align*} -{\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}-{b\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}-2\,{\frac{\sqrt{2}}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }-{\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}+{b\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}+2\,{\frac{\sqrt{2}}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.08545, size = 1226, normalized size = 7.66 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} + \sqrt{b^{2} - 4}}{b^{2} - 4}} \log \left (\frac{1}{2} \,{\left (2 \, b + 3 \, \sqrt{2}\right )} x + \frac{1}{2} \, \sqrt{\frac{1}{2}}{\left (b^{2} - \frac{b^{3} + \sqrt{2} b^{2} - 4 \, b - 4 \, \sqrt{2}}{\sqrt{b^{2} - 4}} - 4\right )} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} + \sqrt{b^{2} - 4}}{b^{2} - 4}}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} + \sqrt{b^{2} - 4}}{b^{2} - 4}} \log \left (\frac{1}{2} \,{\left (2 \, b + 3 \, \sqrt{2}\right )} x - \frac{1}{2} \, \sqrt{\frac{1}{2}}{\left (b^{2} - \frac{b^{3} + \sqrt{2} b^{2} - 4 \, b - 4 \, \sqrt{2}}{\sqrt{b^{2} - 4}} - 4\right )} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} + \sqrt{b^{2} - 4}}{b^{2} - 4}}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} - \sqrt{b^{2} - 4}}{b^{2} - 4}} \log \left (\frac{1}{2} \,{\left (2 \, b + 3 \, \sqrt{2}\right )} x + \frac{1}{2} \, \sqrt{\frac{1}{2}}{\left (b^{2} + \frac{b^{3} + \sqrt{2} b^{2} - 4 \, b - 4 \, \sqrt{2}}{\sqrt{b^{2} - 4}} - 4\right )} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} - \sqrt{b^{2} - 4}}{b^{2} - 4}}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} - \sqrt{b^{2} - 4}}{b^{2} - 4}} \log \left (\frac{1}{2} \,{\left (2 \, b + 3 \, \sqrt{2}\right )} x - \frac{1}{2} \, \sqrt{\frac{1}{2}}{\left (b^{2} + \frac{b^{3} + \sqrt{2} b^{2} - 4 \, b - 4 \, \sqrt{2}}{\sqrt{b^{2} - 4}} - 4\right )} \sqrt{-\frac{3 \, b + 4 \, \sqrt{2} - \sqrt{b^{2} - 4}}{b^{2} - 4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(1/2)*sqrt(-(3*b + 4*sqrt(2) + sqrt(b^2 - 4))/(b^2 - 4))*log(1/2*(2*b + 3*sqrt(2))*x + 1/2*sqrt(1/2)*
(b^2 - (b^3 + sqrt(2)*b^2 - 4*b - 4*sqrt(2))/sqrt(b^2 - 4) - 4)*sqrt(-(3*b + 4*sqrt(2) + sqrt(b^2 - 4))/(b^2 -
4))) + 1/2*sqrt(1/2)*sqrt(-(3*b + 4*sqrt(2) + sqrt(b^2 - 4))/(b^2 - 4))*log(1/2*(2*b + 3*sqrt(2))*x - 1/2*sqr
t(1/2)*(b^2 - (b^3 + sqrt(2)*b^2 - 4*b - 4*sqrt(2))/sqrt(b^2 - 4) - 4)*sqrt(-(3*b + 4*sqrt(2) + sqrt(b^2 - 4))
/(b^2 - 4))) - 1/2*sqrt(1/2)*sqrt(-(3*b + 4*sqrt(2) - sqrt(b^2 - 4))/(b^2 - 4))*log(1/2*(2*b + 3*sqrt(2))*x +
1/2*sqrt(1/2)*(b^2 + (b^3 + sqrt(2)*b^2 - 4*b - 4*sqrt(2))/sqrt(b^2 - 4) - 4)*sqrt(-(3*b + 4*sqrt(2) - sqrt(b^
2 - 4))/(b^2 - 4))) + 1/2*sqrt(1/2)*sqrt(-(3*b + 4*sqrt(2) - sqrt(b^2 - 4))/(b^2 - 4))*log(1/2*(2*b + 3*sqrt(2
))*x - 1/2*sqrt(1/2)*(b^2 + (b^3 + sqrt(2)*b^2 - 4*b - 4*sqrt(2))/sqrt(b^2 - 4) - 4)*sqrt(-(3*b + 4*sqrt(2) -
sqrt(b^2 - 4))/(b^2 - 4)))

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Sympy [B]  time = 1.80991, size = 332, normalized size = 2.08 \begin{align*} - \operatorname{RootSum}{\left (t^{4} \left (16 b^{4} - 128 b^{2} + 256\right ) + t^{2} \left (12 b^{3} + 16 \sqrt{2} b^{2} - 48 b - 64 \sqrt{2}\right ) + 2 b^{2} + 6 \sqrt{2} b + 9, \left ( t \mapsto t \log{\left (\frac{t^{3} \left (64 b^{12} + 672 \sqrt{2} b^{11} + 5760 b^{10} + 12064 \sqrt{2} b^{9} + 17744 b^{8} - 27480 \sqrt{2} b^{7} - 154608 b^{6} - 141376 \sqrt{2} b^{5} - 69072 b^{4} + 61704 \sqrt{2} b^{3} + 78192 b^{2} - 2592 \sqrt{2} b - 15552\right )}{8 b^{10} + 88 \sqrt{2} b^{9} + 828 b^{8} + 2144 \sqrt{2} b^{7} + 6470 b^{6} + 5310 \sqrt{2} b^{5} + 2781 b^{4} - 2322 \sqrt{2} b^{3} - 3402 b^{2} + 729} + \frac{t \left (16 b^{7} + 116 \sqrt{2} b^{6} + 668 b^{5} + 942 \sqrt{2} b^{4} + 1226 b^{3} + 144 \sqrt{2} b^{2} - 378 b - 108 \sqrt{2}\right )}{4 b^{6} + 28 \sqrt{2} b^{5} + 152 b^{4} + 192 \sqrt{2} b^{3} + 189 b^{2} - 27 \sqrt{2} b - 81} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(_t**4*(16*b**4 - 128*b**2 + 256) + _t**2*(12*b**3 + 16*sqrt(2)*b**2 - 48*b - 64*sqrt(2)) + 2*b**2 + 6
*sqrt(2)*b + 9, Lambda(_t, _t*log(_t**3*(64*b**12 + 672*sqrt(2)*b**11 + 5760*b**10 + 12064*sqrt(2)*b**9 + 1774
4*b**8 - 27480*sqrt(2)*b**7 - 154608*b**6 - 141376*sqrt(2)*b**5 - 69072*b**4 + 61704*sqrt(2)*b**3 + 78192*b**2
- 2592*sqrt(2)*b - 15552)/(8*b**10 + 88*sqrt(2)*b**9 + 828*b**8 + 2144*sqrt(2)*b**7 + 6470*b**6 + 5310*sqrt(2
)*b**5 + 2781*b**4 - 2322*sqrt(2)*b**3 - 3402*b**2 + 729) + _t*(16*b**7 + 116*sqrt(2)*b**6 + 668*b**5 + 942*sq
rt(2)*b**4 + 1226*b**3 + 144*sqrt(2)*b**2 - 378*b - 108*sqrt(2))/(4*b**6 + 28*sqrt(2)*b**5 + 152*b**4 + 192*sq
rt(2)*b**3 + 189*b**2 - 27*sqrt(2)*b - 81) + x)))

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Giac [C]  time = 1.25828, size = 2847, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*cosh(1/2*imag_part(arcsin(1/2
*b)))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b))) - (b^2 + sqrt(b^2 - 4)*b - 4)*cosh(1/2*imag_part(arcsin(1/2
*b)))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^3 - 9*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_par
t(arcsin(1/2*b)))^2*cosh(1/2*imag_part(arcsin(1/2*b)))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*sinh(1/2*i
mag_part(arcsin(1/2*b))) + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cosh(1/2*imag_part(arcsin(1/2*b)))^2*sin(5/4*pi + 1/2
*real_part(arcsin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b))) + 9*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1
/2*real_part(arcsin(1/2*b)))^2*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*s
inh(1/2*imag_part(arcsin(1/2*b)))^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(5/4
*pi + 1/2*real_part(arcsin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos
(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin
(1/2*b)))^3 + (b^2 + sqrt(b^2 - 4)*b - 4)*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*sinh(1/2*imag_part(arcs
in(1/2*b)))^3 - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(5/4
*pi + 1/2*real_part(arcsin(1/2*b))) + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*sin(5/4*pi + 1/2*rea
l_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b))))*arctan((x - cos(5/4*pi + 1/2*arcsin(1/2*b)))/sin(5/
4*pi + 1/2*arcsin(1/2*b)))/(b^2 - 4) - 1/2*(3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/
2*b)))^2*cosh(1/2*imag_part(arcsin(1/2*b)))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b))) - (b^2 + sqrt(b^2 - 4
)*b - 4)*cosh(1/2*imag_part(arcsin(1/2*b)))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^3 - 9*(b^2 + sqrt(b^2
- 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*cosh(1/2*imag_part(arcsin(1/2*b)))^2*sin(1/4*pi + 1/
2*real_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b))) + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cosh(1/2*imag_p
art(arcsin(1/2*b)))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b))) + 9*(b^2
+ sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(1/
4*pi + 1/2*real_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cosh
(1/2*imag_part(arcsin(1/2*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b)))
^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*sin(1/4*pi + 1/2*real_part(arc
sin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b)))^3 + (b^2 + sqrt(b^2 - 4)*b - 4)*sin(1/4*pi + 1/2*real_part(arc
sin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b)))^3 - (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cosh
(1/2*imag_part(arcsin(1/2*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b))) + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 -
4)*b - 4*sqrt(2))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b))))*arctan((x - c
os(1/4*pi + 1/2*arcsin(1/2*b)))/sin(1/4*pi + 1/2*arcsin(1/2*b)))/(b^2 - 4) + 1/4*((b^2 + sqrt(b^2 - 4)*b - 4)*
cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*cosh(1/2*imag_part(arcsin(1/2*b)))^3 - 3*(b^2 + sqrt(b^2 - 4)*b -
4)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))^3*sin(5/4*pi + 1/2*real_part
(arcsin(1/2*b)))^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*cosh(1/2*imag_
part(arcsin(1/2*b)))^2*sinh(1/2*imag_part(arcsin(1/2*b))) + 9*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*rea
l_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*sinh(
1/2*imag_part(arcsin(1/2*b))) + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*cos
h(1/2*imag_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - 9*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*p
i + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b))
)^2*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - (b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b
)))^3*sinh(1/2*imag_part(arcsin(1/2*b)))^3 + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(5/4*pi + 1/2*real_part(arcsin(1
/2*b)))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*sinh(1/2*imag_part(arcsin(1/2*b)))^3 - (sqrt(2)*b^2 + sqr
t(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b))
) + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b)))*sinh(1/2*im
ag_part(arcsin(1/2*b))))*log(x^2 - 2*x*cos(5/4*pi + 1/2*arcsin(1/2*b)) + 1)/(b^2 - 4) + 1/4*((b^2 + sqrt(b^2 -
4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*cosh(1/2*imag_part(arcsin(1/2*b)))^3 - 3*(b^2 + sqrt(b
^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))^3*sin(1/4*pi + 1/
2*real_part(arcsin(1/2*b)))^2 - 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^3*cos
h(1/2*imag_part(arcsin(1/2*b)))^2*sinh(1/2*imag_part(arcsin(1/2*b))) + 9*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*p
i + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b
)))^2*sinh(1/2*imag_part(arcsin(1/2*b))) + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(arcsin(1/2
*b)))^3*cosh(1/2*imag_part(arcsin(1/2*b)))*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - 9*(b^2 + sqrt(b^2 - 4)*b - 4
)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arcsin(1/2*b)))*sin(1/4*pi + 1/2*real_part(arc
sin(1/2*b)))^2*sinh(1/2*imag_part(arcsin(1/2*b)))^2 - (b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_part(a
rcsin(1/2*b)))^3*sinh(1/2*imag_part(arcsin(1/2*b)))^3 + 3*(b^2 + sqrt(b^2 - 4)*b - 4)*cos(1/4*pi + 1/2*real_pa
rt(arcsin(1/2*b)))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b)))^2*sinh(1/2*imag_part(arcsin(1/2*b)))^3 - (sqrt(2
)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))*cosh(1/2*imag_part(arc
sin(1/2*b))) + (sqrt(2)*b^2 + sqrt(2)*sqrt(b^2 - 4)*b - 4*sqrt(2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b)))*
sinh(1/2*imag_part(arcsin(1/2*b))))*log(x^2 - 2*x*cos(1/4*pi + 1/2*arcsin(1/2*b)) + 1)/(b^2 - 4)