3.1.82 \(\int \frac {1}{1+\cos ^8(x)} \, dx\) [82]

Optimal. Leaf size=129 \[ -\frac {\text {ArcTan}\left (\sqrt {1-\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}-\frac {\text {ArcTan}\left (\sqrt {1+\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}-\frac {\text {ArcTan}\left (\sqrt {1-(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}-\frac {\text {ArcTan}\left (\sqrt {1+(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1+(-1)^{3/4}}} \]

[Out]

-1/4*arctan(cot(x)*(1-(-1)^(1/4))^(1/2))/(1-(-1)^(1/4))^(1/2)-1/4*arctan(cot(x)*(1+(-1)^(1/4))^(1/2))/(1+(-1)^
(1/4))^(1/2)-1/4*arctan(cot(x)*(1-(-1)^(3/4))^(1/2))/(1-(-1)^(3/4))^(1/2)-1/4*arctan(cot(x)*(1+(-1)^(3/4))^(1/
2))/(1+(-1)^(3/4))^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3290, 3260, 209} \begin {gather*} -\frac {\text {ArcTan}\left (\sqrt {1-\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}-\frac {\text {ArcTan}\left (\sqrt {1+\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}-\frac {\text {ArcTan}\left (\sqrt {1-(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}-\frac {\text {ArcTan}\left (\sqrt {1+(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1+(-1)^{3/4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^8)^(-1),x]

[Out]

-1/4*ArcTan[Sqrt[1 - (-1)^(1/4)]*Cot[x]]/Sqrt[1 - (-1)^(1/4)] - ArcTan[Sqrt[1 + (-1)^(1/4)]*Cot[x]]/(4*Sqrt[1
+ (-1)^(1/4)]) - ArcTan[Sqrt[1 - (-1)^(3/4)]*Cot[x]]/(4*Sqrt[1 - (-1)^(3/4)]) - ArcTan[Sqrt[1 + (-1)^(3/4)]*Co
t[x]]/(4*Sqrt[1 + (-1)^(3/4)])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rule 3290

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {1}{1+\cos ^8(x)} \, dx &=\frac {1}{4} \int \frac {1}{1-\sqrt [4]{-1} \cos ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+\sqrt [4]{-1} \cos ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1-(-1)^{3/4} \cos ^2(x)} \, dx+\frac {1}{4} \int \frac {1}{1+(-1)^{3/4} \cos ^2(x)} \, dx\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\left (1-\sqrt [4]{-1}\right ) x^2} \, dx,x,\cot (x)\right )\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\left (1+\sqrt [4]{-1}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\left (1-(-1)^{3/4}\right ) x^2} \, dx,x,\cot (x)\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\left (1+(-1)^{3/4}\right ) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {1-\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt [4]{-1}} \cot (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}}}-\frac {\tan ^{-1}\left (\sqrt {1-(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1-(-1)^{3/4}}}-\frac {\tan ^{-1}\left (\sqrt {1+(-1)^{3/4}} \cot (x)\right )}{4 \sqrt {1+(-1)^{3/4}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.16, size = 141, normalized size = 1.09 \begin {gather*} 8 \text {RootSum}\left [1+8 \text {$\#$1}+28 \text {$\#$1}^2+56 \text {$\#$1}^3+326 \text {$\#$1}^4+56 \text {$\#$1}^5+28 \text {$\#$1}^6+8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \text {ArcTan}\left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{1+7 \text {$\#$1}+21 \text {$\#$1}^2+163 \text {$\#$1}^3+35 \text {$\#$1}^4+21 \text {$\#$1}^5+7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^8)^(-1),x]

[Out]

8*RootSum[1 + 8*#1 + 28*#1^2 + 56*#1^3 + 326*#1^4 + 56*#1^5 + 28*#1^6 + 8*#1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(
Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(1 + 7*#1 + 21*#1^2 + 163*#1^3 + 35*#1^4 + 21*#1^
5 + 7*#1^6 + #1^7) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.35, size = 67, normalized size = 0.52

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}\right )}{8}\)	\(67\)
risch	\(\left (\munderset {\textit {\_R} =\RootOf \left (8192 \textit {\_Z}^{4}+\left (128-128 i\right ) \textit {\_Z}^{2}+1-i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (1024+1024 i\right ) \textit {\_R}^{3}+\left (-128+128 i\right ) \textit {\_R}^{2}+\left (16-16 i\right ) \textit {\_R} +1+2 i\right )\right )+\left (\munderset {\textit {\_R} =\RootOf \left (8192 \textit {\_Z}^{4}+\left (128+128 i\right ) \textit {\_Z}^{2}+1+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-1024+1024 i\right ) \textit {\_R}^{3}+\left (-128-128 i\right ) \textit {\_R}^{2}+\left (-16-16 i\right ) \textit {\_R} +1-2 i\right )\right )\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)^8),x,method=_RETURNVERBOSE)

[Out]

1/8*sum((_R^6+3*_R^4+3*_R^2+1)/(_R^7+3*_R^5+3*_R^3+_R)*ln(tan(x)-_R),_R=RootOf(_Z^8+4*_Z^6+6*_Z^4+4*_Z^2+2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^8),x, algorithm="maxima")

[Out]

integrate(1/(cos(x)^8 + 1), x)

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Fricas [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/(1+cos(x)^8),x, algorithm="fricas")

[Out]

Timed out

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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/(1+cos(x)**8),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^8),x, algorithm="giac")

[Out]

sage0*x

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Mupad [B]
time = 3.11, size = 1025, normalized size = 7.95 \begin {gather*} -\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\right )\,\sqrt {-\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}-1}-\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}-1}+\frac {\mathrm {tan}\left (x\right )\,\sqrt {2\,\sqrt {2}-3}\,\sqrt {-\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}-1}-\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {2\,\sqrt {2}-3}\,\sqrt {-\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}-1}\right )\,\sqrt {-\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}+1}-\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}+1}-\frac {\mathrm {tan}\left (x\right )\,\sqrt {2\,\sqrt {2}-3}\,\sqrt {\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}+1}+\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {2\,\sqrt {2}-3}\,\sqrt {\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}-\sqrt {2\,\sqrt {2}-3}+1}\right )\,\sqrt {\frac {\sqrt {2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\right )\,\sqrt {-\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}+1}+\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}+1}+\frac {\mathrm {tan}\left (x\right )\,\sqrt {-2\,\sqrt {2}-3}\,\sqrt {-\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}+1}+\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {-2\,\sqrt {2}-3}\,\sqrt {-\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}+\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}+1}\right )\,\sqrt {-\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}-1}+\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}-1}-\frac {\mathrm {tan}\left (x\right )\,\sqrt {-2\,\sqrt {2}-3}\,\sqrt {\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,8{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}-1}-\frac {\sqrt {2}\,\mathrm {tan}\left (x\right )\,\sqrt {-2\,\sqrt {2}-3}\,\sqrt {\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,4{}\mathrm {i}}{\frac {\sqrt {2}\,\sqrt {-2\,\sqrt {2}-3}}{2}-\frac {\sqrt {2}}{2}+\sqrt {-2\,\sqrt {2}-3}-1}\right )\,\sqrt {\frac {\sqrt {-2\,\sqrt {2}-3}}{128}-\frac {1}{128}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^8 + 1),x)

[Out]

atan((tan(x)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2
*2^(1/2) - 3)^(1/2) + 1) - (2^(1/2)*tan(x)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2)
- 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1) - (tan(x)*(2*2^(1/2) - 3)^(1/2)*((2*2^(1/2) - 3)^(1/2)/
128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1) + (2^(1/2)*
tan(x)*(2*2^(1/2) - 3)^(1/2)*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2
- 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) + 1))*((2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*2i - atan((tan(x)*(- (2*2^
(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2)
- 1) - (2^(1/2)*tan(x)*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2
^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1) + (tan(x)*(2*2^(1/2) - 3)^(1/2)*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1
/2)*8i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 - (2*2^(1/2) - 3)^(1/2) - 1) - (2^(1/2)*tan(x)*(2*2^(1/
2) - 3)^(1/2)*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 -
(2*2^(1/2) - 3)^(1/2) - 1))*(- (2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*2i + atan((tan(x)*(- (- 2*2^(1/2) - 3
)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) + 1)
+ (2^(1/2)*tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 + 2^
(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) + 1) + (tan(x)*(- 2*2^(1/2) - 3)^(1/2)*(- (- 2*2^(1/2) - 3)^(1/2)/128 - 1/12
8)^(1/2)*8i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 + 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) + 1) + (2^(1/2)*tan(x)
*(- 2*2^(1/2) - 3)^(1/2)*(- (- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/
2 + 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) + 1))*(- (- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*2i - atan((tan(x)*
((- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 + (- 2*2^(1/2
) - 3)^(1/2) - 1) + (2^(1/2)*tan(x)*((- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(- 2*2^(1/2) - 3
)^(1/2))/2 - 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) - 1) - (tan(x)*(- 2*2^(1/2) - 3)^(1/2)*((- 2*2^(1/2) - 3)^(1/
2)/128 - 1/128)^(1/2)*8i)/((2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/2 - 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) - 1) - (2
^(1/2)*tan(x)*(- 2*2^(1/2) - 3)^(1/2)*((- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*4i)/((2^(1/2)*(- 2*2^(1/2) -
3)^(1/2))/2 - 2^(1/2)/2 + (- 2*2^(1/2) - 3)^(1/2) - 1))*((- 2*2^(1/2) - 3)^(1/2)/128 - 1/128)^(1/2)*2i

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