3.5.1 \(\int \frac {1}{x^2 (1-3 x^4+x^8)} \, dx\) [401]

Optimal. Leaf size=172 \[ -\frac {1}{x}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}} \]

[Out]

-1/x+1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(984-440*5^(1/2))^(1/4)*5^(1/2)-1/20*arctanh(2^(1/4)*x*(1/(3
+5^(1/2)))^(1/4))*(984-440*5^(1/2))^(1/4)*5^(1/2)-1/40*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(3+5^(1/2))^(5/
4)*2^(3/4)*5^(1/2)+1/40*arctanh(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(3+5^(1/2))^(5/4)*2^(3/4)*5^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 172, 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 = {1382, 1524, 304, 209, 212} \begin {gather*} \frac {\sqrt [4]{984-440 \sqrt {5}} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {1}{x}-\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 - 3*x^4 + x^8)),x]

[Out]

-x^(-1) + ((984 - 440*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(4*Sqrt[5]) - ((3 + Sqrt[5])^(5/4)*Arc
Tan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*Sqrt[5]) - ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/
4)*x])/(4*Sqrt[5]) + ((3 + Sqrt[5])^(5/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*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 304

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

Rule 1382

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

Rule 1524

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (1-3 x^4+x^8\right )} \, dx &=-\frac {1}{x}+\int \frac {x^2 \left (3-x^4\right )}{1-3 x^4+x^8} \, dx\\ &=-\frac {1}{x}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\frac {1}{x}-\frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}\\ &=-\frac {1}{x}+\frac {\sqrt [4]{984-440 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}}-\frac {\sqrt [4]{984-440 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 174, normalized size = 1.01 \begin {gather*} -\frac {1}{x}-\frac {\left (3+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (-3+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\left (-3+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 - 3*x^4 + x^8)),x]

[Out]

-x^(-1) - ((3 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((-3 + Sqrt[5])*ArcTa
n[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) + ((3 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*
Sqrt[10*(-1 + Sqrt[5])]) + ((-3 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])])

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 135, normalized size = 0.78

method	result	size
risch	\(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-20 \textit {\_R}^{3}-47 \textit {\_R} +5 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-20 \textit {\_R}^{3}+47 \textit {\_R} +5 x \right )\right )}{4}\)	\(73\)
default	\(\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\sqrt {5}\, \left (\sqrt {5}-3\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (\sqrt {5}-3\right ) \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {1}{x}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/10*(3+5^(1/2))*5^(1/2)/(2*5^(1/2)-2)^(1/2)*arctanh(2*x/(2*5^(1/2)-2)^(1/2))-1/10*5^(1/2)*(5^(1/2)-3)/(2*5^(1
/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+1/10*5^(1/2)*(5^(1/2)-3)/(2*5^(1/2)+2)^(1/2)*arctanh(2*x/(2*5^(1/
2)+2)^(1/2))-1/10*(3+5^(1/2))*5^(1/2)/(2*5^(1/2)-2)^(1/2)*arctan(2*x/(2*5^(1/2)-2)^(1/2))-1/x

________________________________________________________________________________________

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/x^2/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (118) = 236\).
time = 0.37, size = 321, normalized size = 1.87 \begin {gather*} -\frac {4 \, \sqrt {10} x \sqrt {5 \, \sqrt {5} - 11} \arctan \left (\frac {1}{40} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} + 1} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} - \frac {1}{20} \, \sqrt {10} {\left (3 \, \sqrt {5} x + 5 \, x\right )} \sqrt {5 \, \sqrt {5} - 11}\right ) - 4 \, \sqrt {10} x \sqrt {5 \, \sqrt {5} + 11} \arctan \left (\frac {1}{40} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} - 1} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} - \frac {1}{20} \, \sqrt {10} {\left (3 \, \sqrt {5} x - 5 \, x\right )} \sqrt {5 \, \sqrt {5} + 11}\right ) + \sqrt {10} x \sqrt {5 \, \sqrt {5} - 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (2 \, \sqrt {5} + 5\right )} + 10 \, x\right ) - \sqrt {10} x \sqrt {5 \, \sqrt {5} - 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (2 \, \sqrt {5} + 5\right )} + 10 \, x\right ) + \sqrt {10} x \sqrt {5 \, \sqrt {5} + 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (2 \, \sqrt {5} - 5\right )} + 10 \, x\right ) - \sqrt {10} x \sqrt {5 \, \sqrt {5} + 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (2 \, \sqrt {5} - 5\right )} + 10 \, x\right ) + 40}{40 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/40*(4*sqrt(10)*x*sqrt(5*sqrt(5) - 11)*arctan(1/40*sqrt(10)*sqrt(2)*sqrt(2*x^2 + sqrt(5) + 1)*sqrt(5*sqrt(5)
- 11)*(3*sqrt(5) + 5) - 1/20*sqrt(10)*(3*sqrt(5)*x + 5*x)*sqrt(5*sqrt(5) - 11)) - 4*sqrt(10)*x*sqrt(5*sqrt(5)
+ 11)*arctan(1/40*sqrt(10)*sqrt(2)*sqrt(2*x^2 + sqrt(5) - 1)*sqrt(5*sqrt(5) + 11)*(3*sqrt(5) - 5) - 1/20*sqrt
(10)*(3*sqrt(5)*x - 5*x)*sqrt(5*sqrt(5) + 11)) + sqrt(10)*x*sqrt(5*sqrt(5) - 11)*log(sqrt(10)*sqrt(5*sqrt(5) -
11)*(2*sqrt(5) + 5) + 10*x) - sqrt(10)*x*sqrt(5*sqrt(5) - 11)*log(-sqrt(10)*sqrt(5*sqrt(5) - 11)*(2*sqrt(5) +
5) + 10*x) + sqrt(10)*x*sqrt(5*sqrt(5) + 11)*log(sqrt(10)*sqrt(5*sqrt(5) + 11)*(2*sqrt(5) - 5) + 10*x) - sqrt
(10)*x*sqrt(5*sqrt(5) + 11)*log(-sqrt(10)*sqrt(5*sqrt(5) + 11)*(2*sqrt(5) - 5) + 10*x) + 40)/x

________________________________________________________________________________________

Sympy [A]
time = 0.81, size = 63, normalized size = 0.37 \begin {gather*} \operatorname {RootSum} {\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {19251200 t^{7}}{11} - \frac {369792 t^{3}}{11} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {19251200 t^{7}}{11} - \frac {369792 t^{3}}{11} + x \right )} \right )\right )} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 880*_t**2 - 1, Lambda(_t, _t*log(19251200*_t**7/11 - 369792*_t**3/11 + x))) + RootSum(640
0*_t**4 + 880*_t**2 - 1, Lambda(_t, _t*log(19251200*_t**7/11 - 369792*_t**3/11 + x))) - 1/x

________________________________________________________________________________________

Giac [A]
time = 3.54, size = 152, normalized size = 0.88 \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 ) - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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) - 1
10)*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/x

________________________________________________________________________________________

Mupad [B]
time = 1.34, size = 250, normalized size = 1.45 \begin {gather*} -\frac {1}{x}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-50\,\sqrt {5}-110}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {\sqrt {5}\,x\,\sqrt {-50\,\sqrt {5}-110}\,517{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}\right )\,\sqrt {-50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {110-50\,\sqrt {5}}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {\sqrt {5}\,x\,\sqrt {110-50\,\sqrt {5}}\,517{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,\sqrt {110-50\,\sqrt {5}}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}-110}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}-\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}-110}\,517{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}-6765\right )}\right )\,\sqrt {50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}+110}\,1155{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\right )}+\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}+110}\,517{}\mathrm {i}}{2\,\left (3025\,\sqrt {5}+6765\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(1/(x^2*(x^8 - 3*x^4 + 1)),x)

[Out]

(atan((x*(110 - 50*5^(1/2))^(1/2)*1155i)/(2*(3025*5^(1/2) - 6765)) - (5^(1/2)*x*(110 - 50*5^(1/2))^(1/2)*517i)
/(2*(3025*5^(1/2) - 6765)))*(110 - 50*5^(1/2))^(1/2)*1i)/20 - (atan((x*(- 50*5^(1/2) - 110)^(1/2)*1155i)/(2*(3
025*5^(1/2) + 6765)) + (5^(1/2)*x*(- 50*5^(1/2) - 110)^(1/2)*517i)/(2*(3025*5^(1/2) + 6765)))*(- 50*5^(1/2) -
110)^(1/2)*1i)/20 + (atan((x*(50*5^(1/2) - 110)^(1/2)*1155i)/(2*(3025*5^(1/2) - 6765)) - (5^(1/2)*x*(50*5^(1/2
) - 110)^(1/2)*517i)/(2*(3025*5^(1/2) - 6765)))*(50*5^(1/2) - 110)^(1/2)*1i)/20 - (atan((x*(50*5^(1/2) + 110)^
(1/2)*1155i)/(2*(3025*5^(1/2) + 6765)) + (5^(1/2)*x*(50*5^(1/2) + 110)^(1/2)*517i)/(2*(3025*5^(1/2) + 6765)))*
(50*5^(1/2) + 110)^(1/2)*1i)/20 - 1/x

________________________________________________________________________________________