3.402 \(\int \frac {1}{x^4 (1-3 x^4+x^8)} \, dx\)

Optimal. Leaf size=182 \[ -\frac {1}{3 x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4\ 2^{3/4} \sqrt {5}} \]

[Out]

-1/3/x^3-1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(843-377*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))*(843-377*5^(1/2))^(1/4)*2^(3/4)*5^(1/2)+1/20*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3
/4))*(843+377*5^(1/2))^(1/4)*2^(3/4)*5^(1/2)+1/20*arctanh(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(843+377*5^(1/2))^(
1/4)*2^(3/4)*5^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 182, 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 = {1368, 1422, 212, 206, 203} \[ -\frac {1}{3 x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4\ 2^{3/4} \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(3*x^3) - (((843 - 377*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((3 + Sqrt[5])^(7
/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(3/4)*Sqrt[5]) - (((843 - 377*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sq
rt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((3 + Sqrt[5])^(7/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(3/4)*Sqrt[5])

Rule 203

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

Rule 206

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 1368

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 1422

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 {1}{x^4 \left (1-3 x^4+x^8\right )} \, dx &=-\frac {1}{3 x^3}+\frac {1}{3} \int \frac {9-3 x^4}{1-3 x^4+x^8} \, dx\\ &=-\frac {1}{3 x^3}+\frac {1}{10} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=-\frac {1}{3 x^3}+\frac {\left (5-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}+\frac {\left (5-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}+\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{4 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{3/2} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{4 \sqrt {5}}\\ &=-\frac {1}{3 x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \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 (843+377 \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 (843-377 \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 (843+377 \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 = 0.26, size = 166, normalized size = 0.91 \[ -\frac {1}{3 x^3}+\frac {\left (2+\sqrt {5}\right ) \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}-\frac {\left (\sqrt {5}-2\right ) \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\left (2+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}-\frac {\left (\sqrt {5}-2\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.93, size = 327, normalized size = 1.80 \[ \frac {12 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} + 29} \arctan \left (\frac {1}{20} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (2 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (2 \, \sqrt {5} x - 5 \, x\right )}\right )} \sqrt {13 \, \sqrt {5} + 29}\right ) + 12 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} - 29} \arctan \left (\frac {1}{20} \, {\left (\sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (2 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} - 2 \, \sqrt {10} {\left (2 \, \sqrt {5} x + 5 \, x\right )}\right )} \sqrt {13 \, \sqrt {5} - 29}\right ) - 3 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} - 29} \log \left (\sqrt {10} \sqrt {13 \, \sqrt {5} - 29} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) + 3 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} - 29} \log \left (-\sqrt {10} \sqrt {13 \, \sqrt {5} - 29} {\left (7 \, \sqrt {5} + 15\right )} + 20 \, x\right ) + 3 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} + 29} \log \left (\sqrt {10} \sqrt {13 \, \sqrt {5} + 29} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) - 3 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} + 29} \log \left (-\sqrt {10} \sqrt {13 \, \sqrt {5} + 29} {\left (7 \, \sqrt {5} - 15\right )} + 20 \, x\right ) - 40}{120 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(12*sqrt(10)*x^3*sqrt(13*sqrt(5) + 29)*arctan(1/20*(sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1)*(2*sqrt(5)*sqrt(2
) - 5*sqrt(2)) - 2*sqrt(10)*(2*sqrt(5)*x - 5*x))*sqrt(13*sqrt(5) + 29)) + 12*sqrt(10)*x^3*sqrt(13*sqrt(5) - 29
)*arctan(1/20*(sqrt(10)*sqrt(2*x^2 + sqrt(5) + 1)*(2*sqrt(5)*sqrt(2) + 5*sqrt(2)) - 2*sqrt(10)*(2*sqrt(5)*x +
5*x))*sqrt(13*sqrt(5) - 29)) - 3*sqrt(10)*x^3*sqrt(13*sqrt(5) - 29)*log(sqrt(10)*sqrt(13*sqrt(5) - 29)*(7*sqrt
(5) + 15) + 20*x) + 3*sqrt(10)*x^3*sqrt(13*sqrt(5) - 29)*log(-sqrt(10)*sqrt(13*sqrt(5) - 29)*(7*sqrt(5) + 15)
+ 20*x) + 3*sqrt(10)*x^3*sqrt(13*sqrt(5) + 29)*log(sqrt(10)*sqrt(13*sqrt(5) + 29)*(7*sqrt(5) - 15) + 20*x) - 3
*sqrt(10)*x^3*sqrt(13*sqrt(5) + 29)*log(-sqrt(10)*sqrt(13*sqrt(5) + 29)*(7*sqrt(5) - 15) + 20*x) - 40)/x^3

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giac [A]  time = 0.67, size = 152, normalized size = 0.84 \[ -\frac {1}{20} \, \sqrt {130 \, \sqrt {5} - 290} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {130 \, \sqrt {5} + 290} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {130 \, \sqrt {5} - 290} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {130 \, \sqrt {5} - 290} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {130 \, \sqrt {5} + 290} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {130 \, \sqrt {5} + 290} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(130*sqrt(5) - 290)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(130*sqrt(5) + 290)*arctan(x/sqrt(1
/2*sqrt(5) - 1/2)) - 1/40*sqrt(130*sqrt(5) - 290)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(130*sqrt(5
) - 290)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(130*sqrt(5) + 290)*log(abs(x + sqrt(1/2*sqrt(5) - 1
/2))) - 1/40*sqrt(130*sqrt(5) + 290)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/3/x^3

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maple [A]  time = 0.03, size = 209, normalized size = 1.15 \[ \frac {2 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}+\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}+\frac {2 \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}+\frac {2 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}+\frac {2 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}-\frac {1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3 \, x^{3}} - \frac {1}{2} \, \int \frac {2 \, x^{2} + 3}{x^{4} + x^{2} - 1}\,{d x} + \frac {1}{2} \, \int \frac {2 \, x^{2} - 3}{x^{4} - x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 268, normalized size = 1.47 \[ \frac {\mathrm {atan}\left (\frac {x\,\sqrt {-130\,\sqrt {5}-290}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}+196417\right )}+\frac {\sqrt {5}\,x\,\sqrt {-130\,\sqrt {5}-290}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}+196417\right )}\right )\,\sqrt {-130\,\sqrt {5}-290}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {290-130\,\sqrt {5}}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}-196417\right )}-\frac {\sqrt {5}\,x\,\sqrt {290-130\,\sqrt {5}}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}-196417\right )}\right )\,\sqrt {290-130\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {1}{3\,x^3}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}-29}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}-196417\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}-29}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}-196417\right )}\right )\,\sqrt {13\,\sqrt {5}-29}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}+29}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}+196417\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}+29}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}+196417\right )}\right )\,\sqrt {13\,\sqrt {5}+29}\,1{}\mathrm {i}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(x^8 - 3*x^4 + 1)),x)

[Out]

(atan((x*(- 130*5^(1/2) - 290)^(1/2)*20735i)/(2*(87841*5^(1/2) + 196417)) + (5^(1/2)*x*(- 130*5^(1/2) - 290)^(
1/2)*46371i)/(10*(87841*5^(1/2) + 196417)))*(- 130*5^(1/2) - 290)^(1/2)*1i)/20 + (atan((x*(290 - 130*5^(1/2))^
(1/2)*20735i)/(2*(87841*5^(1/2) - 196417)) - (5^(1/2)*x*(290 - 130*5^(1/2))^(1/2)*46371i)/(10*(87841*5^(1/2) -
196417)))*(290 - 130*5^(1/2))^(1/2)*1i)/20 - 1/(3*x^3) - (10^(1/2)*atan((10^(1/2)*x*(13*5^(1/2) - 29)^(1/2)*2
0735i)/(2*(87841*5^(1/2) - 196417)) - (5^(1/2)*10^(1/2)*x*(13*5^(1/2) - 29)^(1/2)*46371i)/(10*(87841*5^(1/2) -
196417)))*(13*5^(1/2) - 29)^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(13*5^(1/2) + 29)^(1/2)*20735i)/(2*(878
41*5^(1/2) + 196417)) + (5^(1/2)*10^(1/2)*x*(13*5^(1/2) + 29)^(1/2)*46371i)/(10*(87841*5^(1/2) + 196417)))*(13
*5^(1/2) + 29)^(1/2)*1i)/20

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sympy [A]  time = 1.28, size = 63, normalized size = 0.35 \[ \operatorname {RootSum} {\left (6400 t^{4} - 2320 t^{2} - 1, \left (t \mapsto t \log {\left (\frac {179200 t^{5}}{377} - \frac {23112 t}{377} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 2320 t^{2} - 1, \left (t \mapsto t \log {\left (\frac {179200 t^{5}}{377} - \frac {23112 t}{377} + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(6400*_t**4 - 2320*_t**2 - 1, Lambda(_t, _t*log(179200*_t**5/377 - 23112*_t/377 + x))) + RootSum(6400*_
t**4 + 2320*_t**2 - 1, Lambda(_t, _t*log(179200*_t**5/377 - 23112*_t/377 + x))) - 1/(3*x**3)

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