3.5.3 \(\int \frac {1}{x^6 (1-3 x^4+x^8)} \, dx\) [403]

Optimal. Leaf size=173 \[ -\frac {1}{5 x^5}-\frac {3}{x}+\frac {\sqrt [4]{2889-1292 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{2889+1292 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{2889-1292 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{2889+1292 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]

[Out]

-1/5/x^5-3/x+1/10*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(2889-1292*5^(1/2))^(1/4)*5^(1/2)-1/10*arctanh(2^(1/
4)*x*(1/(3+5^(1/2)))^(1/4))*(2889-1292*5^(1/2))^(1/4)*5^(1/2)-1/10*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(28
89+1292*5^(1/2))^(1/4)*5^(1/2)+1/10*arctanh(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(2889+1292*5^(1/2))^(1/4)*5^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1382, 1518, 1524, 304, 209, 212} \begin {gather*} \frac {\sqrt [4]{2889-1292 \sqrt {5}} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{2889+1292 \sqrt {5}} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {1}{5 x^5}-\frac {3}{x}-\frac {\sqrt [4]{2889-1292 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{2889+1292 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/5*1/x^5 - 3/x + ((2889 - 1292*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) - ((2889 + 1292
*Sqrt[5])^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - ((2889 - 1292*Sqrt[5])^(1/4)*ArcTanh[(2/(3 +
Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((2889 + 1292*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*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 1518

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, 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^6 \left (1-3 x^4+x^8\right )} \, dx &=-\frac {1}{5 x^5}+\frac {1}{5} \int \frac {15-5 x^4}{x^2 \left (1-3 x^4+x^8\right )} \, dx\\ &=-\frac {1}{5 x^5}-\frac {3}{x}-\frac {1}{5} \int \frac {x^2 \left (-40+15 x^4\right )}{1-3 x^4+x^8} \, dx\\ &=-\frac {1}{5 x^5}-\frac {3}{x}-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\frac {1}{5 x^5}-\frac {3}{x}-\frac {\left (7-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (7-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (7+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (7+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}\\ &=-\frac {1}{5 x^5}-\frac {3}{x}+\frac {\sqrt [4]{46224-20672 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{46224+20672 \sqrt {5}} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{46224-20672 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{4 \sqrt {5}}+\frac {\sqrt [4]{46224+20672 \sqrt {5}} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 189, normalized size = 1.09 \begin {gather*} -\frac {1}{5 x^5}-\frac {3}{x}+\frac {\left (-7-3 \sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\left (7-3 \sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (-7-3 \sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (7-3 \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^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/5*1/x^5 - 3/x + ((-7 - 3*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) + ((7 - 3*S
qrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) - ((-7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(-1 + S
qrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((7 - 3*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 +
Sqrt[5])])

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Maple [A]
time = 0.06, size = 148, normalized size = 0.86

method	result	size
risch	\(\frac {-3 x^{4}-\frac {1}{5}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-380 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-55 \textit {\_R}^{3}+843 \textit {\_R} +34 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+380 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-55 \textit {\_R}^{3}-843 \textit {\_R} +34 x \right )\right )}{4}\)	\(79\)
default	\(\frac {\left (7+3 \sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\left (-7+3 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (-7+3 \sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (7+3 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {1}{5 x^{5}}-\frac {3}{x}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/10*(7+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*(-7+3*5^(1/2))*5^(1/2)/(2
*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+1/10*(-7+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*(7+3*5^(1/2))*5^(1/2)/(2*5^(1/2)-2)^(1/2)*arctan(2*x/(2*5^(1/2)-2)^(1/2))-1/5/x^5-3/
x

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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/x^6/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

-1/5*(15*x^4 + 1)/x^5 - 1/2*integrate((3*x^2 + 5)/(x^4 + x^2 - 1), x) - 1/2*integrate((3*x^2 - 5)/(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. 292 vs. \(2 (121) = 242\).
time = 0.38, size = 292, normalized size = 1.69 \begin {gather*} -\frac {4 \, \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} + 38} \arctan \left (-\frac {1}{4} \, {\left (6 \, \sqrt {5} x - \sqrt {4 \, x^{2} + 2 \, \sqrt {5} - 2} {\left (3 \, \sqrt {5} - 7\right )} - 14 \, x\right )} \sqrt {17 \, \sqrt {5} + 38}\right ) + 4 \, \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} - 38} \arctan \left (-\frac {1}{4} \, {\left (6 \, \sqrt {5} x - \sqrt {4 \, x^{2} + 2 \, \sqrt {5} + 2} {\left (3 \, \sqrt {5} + 7\right )} + 14 \, x\right )} \sqrt {17 \, \sqrt {5} - 38}\right ) + \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} - 38} \log \left (\sqrt {17 \, \sqrt {5} - 38} {\left (5 \, \sqrt {5} + 11\right )} + 2 \, x\right ) - \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} - 38} \log \left (-\sqrt {17 \, \sqrt {5} - 38} {\left (5 \, \sqrt {5} + 11\right )} + 2 \, x\right ) - \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} + 38} \log \left (\sqrt {17 \, \sqrt {5} + 38} {\left (5 \, \sqrt {5} - 11\right )} + 2 \, x\right ) + \sqrt {5} x^{5} \sqrt {17 \, \sqrt {5} + 38} \log \left (-\sqrt {17 \, \sqrt {5} + 38} {\left (5 \, \sqrt {5} - 11\right )} + 2 \, x\right ) + 60 \, x^{4} + 4}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/20*(4*sqrt(5)*x^5*sqrt(17*sqrt(5) + 38)*arctan(-1/4*(6*sqrt(5)*x - sqrt(4*x^2 + 2*sqrt(5) - 2)*(3*sqrt(5) -
7) - 14*x)*sqrt(17*sqrt(5) + 38)) + 4*sqrt(5)*x^5*sqrt(17*sqrt(5) - 38)*arctan(-1/4*(6*sqrt(5)*x - sqrt(4*x^2
+ 2*sqrt(5) + 2)*(3*sqrt(5) + 7) + 14*x)*sqrt(17*sqrt(5) - 38)) + sqrt(5)*x^5*sqrt(17*sqrt(5) - 38)*log(sqrt(
17*sqrt(5) - 38)*(5*sqrt(5) + 11) + 2*x) - sqrt(5)*x^5*sqrt(17*sqrt(5) - 38)*log(-sqrt(17*sqrt(5) - 38)*(5*sqr
t(5) + 11) + 2*x) - sqrt(5)*x^5*sqrt(17*sqrt(5) + 38)*log(sqrt(17*sqrt(5) + 38)*(5*sqrt(5) - 11) + 2*x) + sqrt
(5)*x^5*sqrt(17*sqrt(5) + 38)*log(-sqrt(17*sqrt(5) + 38)*(5*sqrt(5) - 11) + 2*x) + 60*x^4 + 4)/x^5

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Sympy [A]
time = 0.75, size = 73, normalized size = 0.42 \begin {gather*} \operatorname {RootSum} {\left (6400 t^{4} - 6080 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {215808000 t^{7}}{323} - \frac {194833880 t^{3}}{323} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 6080 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {215808000 t^{7}}{323} - \frac {194833880 t^{3}}{323} + x \right )} \right )\right )} + \frac {- 15 x^{4} - 1}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 6080*_t**2 - 1, Lambda(_t, _t*log(215808000*_t**7/323 - 194833880*_t**3/323 + x))) + Root
Sum(6400*_t**4 + 6080*_t**2 - 1, Lambda(_t, _t*log(215808000*_t**7/323 - 194833880*_t**3/323 + x))) + (-15*x**
4 - 1)/(5*x**5)

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Giac [A]
time = 3.96, size = 159, normalized size = 0.92 \begin {gather*} \frac {1}{10} \, \sqrt {85 \, \sqrt {5} - 190} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {85 \, \sqrt {5} + 190} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {85 \, \sqrt {5} - 190} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {85 \, \sqrt {5} - 190} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {85 \, \sqrt {5} + 190} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {85 \, \sqrt {5} + 190} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {15 \, x^{4} + 1}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

1/10*sqrt(85*sqrt(5) - 190)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/10*sqrt(85*sqrt(5) + 190)*arctan(x/sqrt(1/2*
sqrt(5) - 1/2)) - 1/20*sqrt(85*sqrt(5) - 190)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) - 1
90)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(85*sqrt(5) + 190)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2)))
- 1/20*sqrt(85*sqrt(5) + 190)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/5*(15*x^4 + 1)/x^5

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Mupad [B]
time = 1.49, size = 257, normalized size = 1.49 \begin {gather*} -\frac {3\,x^4+\frac {1}{5}}{x^5}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-85\,\sqrt {5}-190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}+5702888}+\frac {\sqrt {5}\,x\,\sqrt {-85\,\sqrt {5}-190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}+5702888\right )}\right )\,\sqrt {-85\,\sqrt {5}-190}\,1{}\mathrm {i}}{10}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {190-85\,\sqrt {5}}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}-5702888}-\frac {\sqrt {5}\,x\,\sqrt {190-85\,\sqrt {5}}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}-5702888\right )}\right )\,\sqrt {190-85\,\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {85\,\sqrt {5}-190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}-5702888}-\frac {\sqrt {5}\,x\,\sqrt {85\,\sqrt {5}-190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}-5702888\right )}\right )\,\sqrt {85\,\sqrt {5}-190}\,1{}\mathrm {i}}{10}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {85\,\sqrt {5}+190}\,372096{}\mathrm {i}}{2550408\,\sqrt {5}+5702888}+\frac {\sqrt {5}\,x\,\sqrt {85\,\sqrt {5}+190}\,832048{}\mathrm {i}}{5\,\left (2550408\,\sqrt {5}+5702888\right )}\right )\,\sqrt {85\,\sqrt {5}+190}\,1{}\mathrm {i}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^8 - 3*x^4 + 1)),x)

[Out]

(atan((x*(190 - 85*5^(1/2))^(1/2)*372096i)/(2550408*5^(1/2) - 5702888) - (5^(1/2)*x*(190 - 85*5^(1/2))^(1/2)*8
32048i)/(5*(2550408*5^(1/2) - 5702888)))*(190 - 85*5^(1/2))^(1/2)*1i)/10 - (atan((x*(- 85*5^(1/2) - 190)^(1/2)
*372096i)/(2550408*5^(1/2) + 5702888) + (5^(1/2)*x*(- 85*5^(1/2) - 190)^(1/2)*832048i)/(5*(2550408*5^(1/2) + 5
702888)))*(- 85*5^(1/2) - 190)^(1/2)*1i)/10 + (atan((x*(85*5^(1/2) - 190)^(1/2)*372096i)/(2550408*5^(1/2) - 57
02888) - (5^(1/2)*x*(85*5^(1/2) - 190)^(1/2)*832048i)/(5*(2550408*5^(1/2) - 5702888)))*(85*5^(1/2) - 190)^(1/2
)*1i)/10 - (atan((x*(85*5^(1/2) + 190)^(1/2)*372096i)/(2550408*5^(1/2) + 5702888) + (5^(1/2)*x*(85*5^(1/2) + 1
90)^(1/2)*832048i)/(5*(2550408*5^(1/2) + 5702888)))*(85*5^(1/2) + 190)^(1/2)*1i)/10 - (3*x^4 + 1/5)/x^5

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