3.403 \(\int \frac{1}{x^6 (1-3 x^4+x^8)} \, dx\)

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 + ((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])

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Rubi [A]  time = 0.14026, antiderivative size = 173, 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, 1510, 298, 203, 206} \[ -\frac{1}{5 x^5}-\frac{3}{x}+\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{2889-1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{2889+1292 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(5*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 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 1510

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]

Rule 298

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 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])

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*}

Mathematica [A]  time = 0.293554, size = 189, normalized size = 1.09 \[ -\frac{1}{5 x^5}-\frac{3}{x}+\frac{\left (-7-3 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\left (7-3 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (-7-3 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (7-3 \sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(5*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 = 0.033, size = 216, normalized size = 1.3 \begin{align*} -{\frac{7\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{3}{2\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{7\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{5\,{x}^{5}}}-3\,{x}^{-1}+{\frac{7\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{3}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{7\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{3}{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^6/(x^8-3*x^4+1),x)

[Out]

-7/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))+3/2/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^
(1/2))^(1/2))-7/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-3/2/(-2+2*5^(1/2))^(1/2)*arct
an(2*x/(-2+2*5^(1/2))^(1/2))-1/5/x^5-3/x+7/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+3
/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+7/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(
1/2))^(1/2))-3/2/(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{15 \, x^{4} + 1}{5 \, x^{5}} - \frac{1}{2} \, \int \frac{3 \, x^{2} + 5}{x^{4} + x^{2} - 1}\,{d x} - \frac{1}{2} \, \int \frac{3 \, x^{2} - 5}{x^{4} - x^{2} - 1}\,{d x} \end{align*}

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]  time = 1.94677, size = 913, normalized size = 5.28 \begin{align*} -\frac{4 \, \sqrt{5} x^{5} \sqrt{17 \, \sqrt{5} + 38} \arctan \left (\frac{1}{4} \,{\left (\sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (3 \, \sqrt{5} \sqrt{2} - 7 \, \sqrt{2}\right )} - 6 \, \sqrt{5} x + 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 (\sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (3 \, \sqrt{5} \sqrt{2} + 7 \, \sqrt{2}\right )} - 6 \, \sqrt{5} x - 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{align*}

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*(sqrt(2*x^2 + sqrt(5) - 1)*(3*sqrt(5)*sqrt(2) - 7*sqrt(2
)) - 6*sqrt(5)*x + 14*x)*sqrt(17*sqrt(5) + 38)) + 4*sqrt(5)*x^5*sqrt(17*sqrt(5) - 38)*arctan(1/4*(sqrt(2*x^2 +
sqrt(5) + 1)*(3*sqrt(5)*sqrt(2) + 7*sqrt(2)) - 6*sqrt(5)*x - 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(-sq
rt(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*s
qrt(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) + 6
0*x^4 + 4)/x^5

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Sympy [A]  time = 0.98265, size = 71, normalized size = 0.41 \begin{align*} \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{align*}

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 = 1.22727, size = 215, normalized size = 1.24 \begin{align*} \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{align*}

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