3.5.4 \(\int \frac {1}{x^8 (1-3 x^4+x^8)} \, dx\) [404]

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

[Out]

-1/7/x^7-1/x^3-1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(39603-17711*5^(1/2))^(1/4)*2^(3/4)*5^(1/2)-1/20*a
rctanh(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(39603-17711*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))*(39603+17711*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))*(
39603+17711*5^(1/2))^(1/4)*2^(3/4)*5^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 189, 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, 1436, 218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \text {ArcTan}\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 )} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {1}{7 x^7}-\frac {1}{x^3}-\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 {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 - 3*x^4 + x^8)),x]

[Out]

-1/7*1/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 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 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 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])

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]

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*}

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Mathematica [A]
time = 0.17, size = 189, normalized size = 1.00 \begin {gather*} -\frac {1}{7 x^7}-\frac {1}{x^3}+\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 ) \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}{-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}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/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 = 0.06, size = 148, normalized size = 0.78

method	result	size
risch	\(\frac {-x^{4}-\frac {1}{7}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}\)	\(79\)
default	\(\frac {\left (11+5 \sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (11+5 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {1}{7 x^{7}}-\frac {1}{x^{3}}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^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] Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (133) = 266\).
time = 0.35, size = 324, normalized size = 1.71 \begin {gather*} \frac {28 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \arctan \left (\frac {1}{40} \, {\left (\sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (11 \, \sqrt {5} - 25\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} \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (11 \, \sqrt {5} + 25\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 {gather*}

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)*sqrt(2*x^2 + sqrt(5) - 1)*(11*sqrt
(5) - 25) - 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)*sqrt(2*x^2 + sqrt(5) + 1)*(11*sqrt(5) + 25) - 2*sqrt(10)*(11*sqrt(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 = 0.86, size = 70, normalized size = 0.37 \begin {gather*} \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 {gather*}

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 = 4.04, size = 159, normalized size = 0.84 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.21, size = 291, normalized size = 1.54 \begin {gather*} -\frac {x^4+\frac {1}{7}}{x^7}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}[phone]\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}[phone]\right )}\right )\,\sqrt {-89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {199-89\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}[phone]\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}[phone]\right )}\right )\,\sqrt {89\,\sqrt {5}+199}\,1{}\mathrm {i}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(x^8 - 3*x^4 + 1)),x)

[Out]

(10^(1/2)*atan((10^(1/2)*x*(- 89*5^(1/2) - 199)^(1/2)*6677047i)/(2*(74049691*5^(1/2) + 165580139)) + (5^(1/2)*
10^(1/2)*x*(- 89*5^(1/2) - 199)^(1/2)*14930373i)/(10*(74049691*5^(1/2) + 165580139)))*(- 89*5^(1/2) - 199)^(1/
2)*1i)/20 - (x^4 + 1/7)/x^7 + (10^(1/2)*atan((10^(1/2)*x*(199 - 89*5^(1/2))^(1/2)*6677047i)/(2*(74049691*5^(1/
2) - 165580139)) - (5^(1/2)*10^(1/2)*x*(199 - 89*5^(1/2))^(1/2)*14930373i)/(10*(74049691*5^(1/2) - 165580139))
)*(199 - 89*5^(1/2))^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(89*5^(1/2) - 199)^(1/2)*6677047i)/(2*(74049691
*5^(1/2) - 165580139)) - (5^(1/2)*10^(1/2)*x*(89*5^(1/2) - 199)^(1/2)*14930373i)/(10*(74049691*5^(1/2) - 16558
0139)))*(89*5^(1/2) - 199)^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(89*5^(1/2) + 199)^(1/2)*6677047i)/(2*(74
049691*5^(1/2) + 165580139)) + (5^(1/2)*10^(1/2)*x*(89*5^(1/2) + 199)^(1/2)*14930373i)/(10*(74049691*5^(1/2) +
165580139)))*(89*5^(1/2) + 199)^(1/2)*1i)/20

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