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

Optimal. Leaf size=189 \[ -\frac{1}{x^3}-\frac{1}{7 x^7}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

[Out]

-1/(7*x^7) - x^(-3) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((3
9603 + 17711*Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - (((39603 - 17711*Sqrt[5])/2)^(
1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTanh[((3 + Sqrt[5
])/2)^(1/4)*x])/(2*Sqrt[5])

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Rubi [A]  time = 0.162722, antiderivative size = 189, normalized size of antiderivative = 1., 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 = {1368, 1504, 1422, 212, 206, 203} \[ -\frac{1}{x^3}-\frac{1}{7 x^7}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (39603+17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(7*x^7) - x^(-3) - (((39603 - 17711*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((3
9603 + 17711*Sqrt[5])/2)^(1/4)*ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - (((39603 - 17711*Sqrt[5])/2)^(
1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + (((39603 + 17711*Sqrt[5])/2)^(1/4)*ArcTanh[((3 + Sqrt[5
])/2)^(1/4)*x])/(2*Sqrt[5])

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 1504

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 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])

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 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 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])

Rubi steps

\begin{align*} \int \frac{1}{x^8 \left (1-3 x^4+x^8\right )} \, dx &=-\frac{1}{7 x^7}+\frac{1}{7} \int \frac{21-7 x^4}{x^4 \left (1-3 x^4+x^8\right )} \, dx\\ &=-\frac{1}{7 x^7}-\frac{1}{x^3}-\frac{1}{21} \int \frac{-168+63 x^4}{1-3 x^4+x^8} \, dx\\ &=-\frac{1}{7 x^7}-\frac{1}{x^3}-\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\\ &=-\frac{1}{7 x^7}-\frac{1}{x^3}-\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}}}+\frac{1}{2} \sqrt{\frac{1}{5} \left (123+55 \sqrt{5}\right )} \int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx+\frac{1}{2} \sqrt{\frac{1}{5} \left (123+55 \sqrt{5}\right )} \int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx\\ &=-\frac{1}{7 x^7}-\frac{1}{x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (39603-17711 \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 (39603+17711 \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 (39603-17711 \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 (39603+17711 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.275646, size = 189, normalized size = 1. \[ -\frac{1}{x^3}-\frac{1}{7 x^7}+\frac{\left (11+5 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\left (11-5 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (-11-5 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (5 \sqrt{5}-11\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*x^7) - x^(-3) + ((11 + 5*Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) + ((11 -
5*Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[10*(1 + Sqrt[5])]) - ((-11 - 5*Sqrt[5])*ArcTanh[Sqrt[2/(-
1 + Sqrt[5])]*x])/(2*Sqrt[10*(-1 + Sqrt[5])]) - ((-11 + 5*Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqrt[1
0*(1 + Sqrt[5])])

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Maple [A]  time = 0.036, size = 216, normalized size = 1.1 \begin{align*}{\frac{11\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{5}{2\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{11\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{5}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{7\,{x}^{7}}}-{x}^{-3}+{\frac{11\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{5}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{11\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{5}{2\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))-5/2/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^
(1/2))^(1/2))+11/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+5/2/(-2+2*5^(1/2))^(1/2)*arc
tan(2*x/(-2+2*5^(1/2))^(1/2))-1/7/x^7-1/x^3+11/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2
))+5/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+11/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+
2*5^(1/2))^(1/2))-5/2/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{7 \, x^{4} + 1}{7 \, x^{7}} - \frac{1}{2} \, \int \frac{5 \, x^{2} + 8}{x^{4} + x^{2} - 1}\,{d x} + \frac{1}{2} \, \int \frac{5 \, x^{2} - 8}{x^{4} - x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(7*x^4 + 1)/x^7 - 1/2*integrate((5*x^2 + 8)/(x^4 + x^2 - 1), x) + 1/2*integrate((5*x^2 - 8)/(x^4 - x^2 -
1), x)

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Fricas [B]  time = 1.95542, size = 1077, normalized size = 5.7 \begin{align*} \frac{28 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} + 199} \arctan \left (\frac{1}{40} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (11 \, \sqrt{5} \sqrt{2} - 25 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (11 \, \sqrt{5} x - 25 \, x\right )}\right )} \sqrt{89 \, \sqrt{5} + 199}\right ) + 28 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} - 199} \arctan \left (\frac{1}{40} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (11 \, \sqrt{5} \sqrt{2} + 25 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (11 \, \sqrt{5} x + 25 \, x\right )}\right )} \sqrt{89 \, \sqrt{5} - 199}\right ) - 7 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} - 199} \log \left (\sqrt{10} \sqrt{89 \, \sqrt{5} - 199}{\left (9 \, \sqrt{5} + 20\right )} + 10 \, x\right ) + 7 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} - 199} \log \left (-\sqrt{10} \sqrt{89 \, \sqrt{5} - 199}{\left (9 \, \sqrt{5} + 20\right )} + 10 \, x\right ) + 7 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} + 199} \log \left (\sqrt{10} \sqrt{89 \, \sqrt{5} + 199}{\left (9 \, \sqrt{5} - 20\right )} + 10 \, x\right ) - 7 \, \sqrt{10} x^{7} \sqrt{89 \, \sqrt{5} + 199} \log \left (-\sqrt{10} \sqrt{89 \, \sqrt{5} + 199}{\left (9 \, \sqrt{5} - 20\right )} + 10 \, x\right ) - 280 \, x^{4} - 40}{280 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/280*(28*sqrt(10)*x^7*sqrt(89*sqrt(5) + 199)*arctan(1/40*(sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1)*(11*sqrt(5)*sqrt
(2) - 25*sqrt(2)) - 2*sqrt(10)*(11*sqrt(5)*x - 25*x))*sqrt(89*sqrt(5) + 199)) + 28*sqrt(10)*x^7*sqrt(89*sqrt(5
) - 199)*arctan(1/40*(sqrt(10)*sqrt(2*x^2 + sqrt(5) + 1)*(11*sqrt(5)*sqrt(2) + 25*sqrt(2)) - 2*sqrt(10)*(11*sq
rt(5)*x + 25*x))*sqrt(89*sqrt(5) - 199)) - 7*sqrt(10)*x^7*sqrt(89*sqrt(5) - 199)*log(sqrt(10)*sqrt(89*sqrt(5)
- 199)*(9*sqrt(5) + 20) + 10*x) + 7*sqrt(10)*x^7*sqrt(89*sqrt(5) - 199)*log(-sqrt(10)*sqrt(89*sqrt(5) - 199)*(
9*sqrt(5) + 20) + 10*x) + 7*sqrt(10)*x^7*sqrt(89*sqrt(5) + 199)*log(sqrt(10)*sqrt(89*sqrt(5) + 199)*(9*sqrt(5)
- 20) + 10*x) - 7*sqrt(10)*x^7*sqrt(89*sqrt(5) + 199)*log(-sqrt(10)*sqrt(89*sqrt(5) + 199)*(9*sqrt(5) - 20) +
10*x) - 280*x^4 - 40)/x^7

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Sympy [A]  time = 0.996892, size = 68, normalized size = 0.36 \begin{align*} \operatorname{RootSum}{\left (6400 t^{4} - 15920 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{460800 t^{5}}{17711} - \frac{2842588 t}{17711} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 15920 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{460800 t^{5}}{17711} - \frac{2842588 t}{17711} + x \right )} \right )\right )} - \frac{7 x^{4} + 1}{7 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(6400*_t**4 - 15920*_t**2 - 1, Lambda(_t, _t*log(460800*_t**5/17711 - 2842588*_t/17711 + x))) + RootSum
(6400*_t**4 + 15920*_t**2 - 1, Lambda(_t, _t*log(460800*_t**5/17711 - 2842588*_t/17711 + x))) - (7*x**4 + 1)/(
7*x**7)

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Giac [A]  time = 1.20188, size = 215, normalized size = 1.14 \begin{align*} -\frac{1}{20} \, \sqrt{890 \, \sqrt{5} - 1990} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{890 \, \sqrt{5} + 1990} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{890 \, \sqrt{5} - 1990} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{890 \, \sqrt{5} - 1990} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{890 \, \sqrt{5} + 1990} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{890 \, \sqrt{5} + 1990} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{7 \, x^{4} + 1}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(890*sqrt(5) - 1990)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(890*sqrt(5) + 1990)*arctan(x/sqrt
(1/2*sqrt(5) - 1/2)) - 1/40*sqrt(890*sqrt(5) - 1990)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqr
t(5) - 1990)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(890*sqrt(5) + 1990)*log(abs(x + sqrt(1/2*sqrt(5
) - 1/2))) - 1/40*sqrt(890*sqrt(5) + 1990)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/7*(7*x^4 + 1)/x^7