3.4.96 \(\int \frac {x^8}{1-3 x^4+x^8} \, dx\) [396]

Optimal. Leaf size=170 \[ x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]

[Out]

x+1/20*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(984-440*5^(1/2))^(1/4)*5^(1/2)+1/20*arctanh(1/2*x*(3+5^(1/2))^
(1/4)*2^(3/4))*(984-440*5^(1/2))^(1/4)*5^(1/2)-1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(123+55*5^(1/2))^(
1/4)*2^(3/4)*5^(1/2)-1/20*arctanh(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(123+55*5^(1/2))^(1/4)*2^(3/4)*5^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1381, 1436, 218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}+x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(1 - 3*x^4 + x^8),x]

[Out]

x - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*
ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)
*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

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 212

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

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 1381

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[d^(2*n - 1)*(d*x)^
(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + 2*n*p + 1))), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && IntegerQ[p]

Rule 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rubi steps

\begin {align*} \int \frac {x^8}{1-3 x^4+x^8} \, dx &=x-\int \frac {1-3 x^4}{1-3 x^4+x^8} \, dx\\ &=x-\frac {1}{10} \left (-15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=x+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}\\ &=x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 160, normalized size = 0.94 \begin {gather*} x+\frac {\left (-2+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (2+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\left (-2+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (2+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(1 - 3*x^4 + x^8),x]

[Out]

x + ((-2 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((2 + Sqrt[5])*ArcTan[Sqrt[2/(
1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] + ((-2 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sq
rt[5])] - ((2 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])]

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Maple [A]
time = 0.08, size = 131, normalized size = 0.77

method	result	size
risch	\(x +\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (15 \textit {\_R}^{3}+29 \textit {\_R} +5 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-15 \textit {\_R}^{3}+29 \textit {\_R} +5 x \right )\right )}{4}\)	\(69\)
default	\(x +\frac {\left (-2+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}-\frac {\sqrt {5}\, \left (2+\sqrt {5}\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}-\frac {\sqrt {5}\, \left (2+\sqrt {5}\right ) \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\left (-2+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

x+1/5*(-2+5^(1/2))*5^(1/2)/(2*5^(1/2)-2)^(1/2)*arctanh(2*x/(2*5^(1/2)-2)^(1/2))-1/5*5^(1/2)*(2+5^(1/2))/(2*5^(
1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))-1/5*5^(1/2)*(2+5^(1/2))/(2*5^(1/2)+2)^(1/2)*arctanh(2*x/(2*5^(1/
2)+2)^(1/2))+1/5*(-2+5^(1/2))*5^(1/2)/(2*5^(1/2)-2)^(1/2)*arctan(2*x/(2*5^(1/2)-2)^(1/2))

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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(x^8/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (118) = 236\).
time = 0.36, size = 312, normalized size = 1.84 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \arctan \left (\frac {1}{20} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} - 1} \sqrt {5 \, \sqrt {5} - 11} {\left (2 \, \sqrt {5} + 5\right )} - \frac {1}{10} \, \sqrt {10} {\left (2 \, \sqrt {5} x + 5 \, x\right )} \sqrt {5 \, \sqrt {5} - 11}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \arctan \left (\frac {1}{20} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} + 1} \sqrt {5 \, \sqrt {5} + 11} {\left (2 \, \sqrt {5} - 5\right )} - \frac {1}{10} \, \sqrt {10} {\left (2 \, \sqrt {5} x - 5 \, x\right )} \sqrt {5 \, \sqrt {5} + 11}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/10*sqrt(10)*sqrt(5*sqrt(5) - 11)*arctan(1/20*sqrt(10)*sqrt(2)*sqrt(2*x^2 + sqrt(5) - 1)*sqrt(5*sqrt(5) - 11
)*(2*sqrt(5) + 5) - 1/10*sqrt(10)*(2*sqrt(5)*x + 5*x)*sqrt(5*sqrt(5) - 11)) - 1/10*sqrt(10)*sqrt(5*sqrt(5) + 1
1)*arctan(1/20*sqrt(10)*sqrt(2)*sqrt(2*x^2 + sqrt(5) + 1)*sqrt(5*sqrt(5) + 11)*(2*sqrt(5) - 5) - 1/10*sqrt(10)
*(2*sqrt(5)*x - 5*x)*sqrt(5*sqrt(5) + 11)) + 1/40*sqrt(10)*sqrt(5*sqrt(5) - 11)*log(sqrt(10)*sqrt(5*sqrt(5) -
11)*(3*sqrt(5) + 5) + 20*x) - 1/40*sqrt(10)*sqrt(5*sqrt(5) - 11)*log(-sqrt(10)*sqrt(5*sqrt(5) - 11)*(3*sqrt(5)
+ 5) + 20*x) - 1/40*sqrt(10)*sqrt(5*sqrt(5) + 11)*log(sqrt(10)*sqrt(5*sqrt(5) + 11)*(3*sqrt(5) - 5) + 20*x) +
1/40*sqrt(10)*sqrt(5*sqrt(5) + 11)*log(-sqrt(10)*sqrt(5*sqrt(5) + 11)*(3*sqrt(5) - 5) + 20*x) + x

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Sympy [A]
time = 0.76, size = 58, normalized size = 0.34 \begin {gather*} x + \operatorname {RootSum} {\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**8-3*x**4+1),x)

[Out]

x + RootSum(6400*_t**4 - 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t**5/11 + 1288*_t/55 + x))) + RootSum(6400*_
t**4 + 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t**5/11 + 1288*_t/55 + x)))

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Giac [A]
time = 4.18, size = 148, normalized size = 0.87 \begin {gather*} -\frac {1}{20} \, \sqrt {50 \, \sqrt {5} + 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

-1/20*sqrt(50*sqrt(5) + 110)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(50*sqrt(5) - 110)*arctan(x/sqrt(1/2
*sqrt(5) - 1/2)) - 1/40*sqrt(50*sqrt(5) + 110)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) +
110)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) - 110)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2)))
- 1/40*sqrt(50*sqrt(5) - 110)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) + x

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Mupad [B]
time = 1.44, size = 246, normalized size = 1.45 \begin {gather*} x-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {-50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {-50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {110-50\,\sqrt {5}}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {110-50\,\sqrt {5}}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {110-50\,\sqrt {5}}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}+110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}+110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {50\,\sqrt {5}+110}\,1{}\mathrm {i}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^8 - 3*x^4 + 1),x)

[Out]

x - (atan((x*(- 50*5^(1/2) - 110)^(1/2)*55i)/(2*(275*5^(1/2) + 605)) + (5^(1/2)*x*(- 50*5^(1/2) - 110)^(1/2)*3
3i)/(2*(275*5^(1/2) + 605)))*(- 50*5^(1/2) - 110)^(1/2)*1i)/20 - (atan((x*(110 - 50*5^(1/2))^(1/2)*55i)/(2*(27
5*5^(1/2) - 605)) - (5^(1/2)*x*(110 - 50*5^(1/2))^(1/2)*33i)/(2*(275*5^(1/2) - 605)))*(110 - 50*5^(1/2))^(1/2)
*1i)/20 + (atan((x*(50*5^(1/2) - 110)^(1/2)*55i)/(2*(275*5^(1/2) - 605)) - (5^(1/2)*x*(50*5^(1/2) - 110)^(1/2)
*33i)/(2*(275*5^(1/2) - 605)))*(50*5^(1/2) - 110)^(1/2)*1i)/20 + (atan((x*(50*5^(1/2) + 110)^(1/2)*55i)/(2*(27
5*5^(1/2) + 605)) + (5^(1/2)*x*(50*5^(1/2) + 110)^(1/2)*33i)/(2*(275*5^(1/2) + 605)))*(50*5^(1/2) + 110)^(1/2)
*1i)/20

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