3.401 \(\int \frac{1}{x^2 (1-3 x^4+x^8)} \, dx\)

Optimal. Leaf size=172 \[ -\frac{1}{x}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

[Out]

-x^(-1) + ((984 - 440*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(4*Sqrt[5]) - ((3 + Sqrt[5])^(5/4)*Arc
Tan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*Sqrt[5]) - ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/
4)*x])/(4*Sqrt[5]) + ((3 + Sqrt[5])^(5/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*Sqrt[5])

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Rubi [A]  time = 0.0887916, antiderivative size = 172, normalized size of antiderivative = 1., 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, 1510, 298, 203, 206} \[ -\frac{1}{x}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 - 3*x^4 + x^8)),x]

[Out]

-x^(-1) + ((984 - 440*Sqrt[5])^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(4*Sqrt[5]) - ((3 + Sqrt[5])^(5/4)*Arc
Tan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*Sqrt[5]) - ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/
4)*x])/(4*Sqrt[5]) + ((3 + Sqrt[5])^(5/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*2^(1/4)*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 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^2 \left (1-3 x^4+x^8\right )} \, dx &=-\frac{1}{x}+\int \frac{x^2 \left (3-x^4\right )}{1-3 x^4+x^8} \, dx\\ &=-\frac{1}{x}+\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{x^2}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx-\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{x^2}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=-\frac{1}{x}-\frac{\left (3-\sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2} \, dx}{2 \sqrt{10}}+\frac{\left (3-\sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2} \, dx}{2 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right ) \int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx}{2 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right ) \int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx}{2 \sqrt{10}}\\ &=-\frac{1}{x}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}}-\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{4 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt [4]{2} \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.270837, size = 174, normalized size = 1.01 \[ -\frac{1}{x}-\frac{\left (3+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (\sqrt{5}-3\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\left (3+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{2 \sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\left (\sqrt{5}-3\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 - 3*x^4 + x^8)),x]

[Out]

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

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Maple [A]  time = 0.024, size = 211, normalized size = 1.2 \begin{align*}{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{3\,\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8-3*x^4+1),x)

[Out]

1/2/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))-3/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(
1/2))^(1/2))-1/2/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-3/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arcta
n(2*x/(-2+2*5^(1/2))^(1/2))+1/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+3/10*5^(1/2)/(-2+2*5^(1
/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))-1/2/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))+3/10*5^(1
/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))-1/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.93412, size = 981, normalized size = 5.7 \begin{align*} \frac{4 \, \sqrt{10} x \sqrt{5 \, \sqrt{5} + 11} \arctan \left (\frac{1}{40} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (3 \, \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (3 \, \sqrt{5} x - 5 \, x\right )}\right )} \sqrt{5 \, \sqrt{5} + 11}\right ) - 4 \, \sqrt{10} x \sqrt{5 \, \sqrt{5} - 11} \arctan \left (\frac{1}{40} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (3 \, \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (3 \, \sqrt{5} x + 5 \, x\right )}\right )} \sqrt{5 \, \sqrt{5} - 11}\right ) - \sqrt{10} x \sqrt{5 \, \sqrt{5} - 11} \log \left (\sqrt{10} \sqrt{5 \, \sqrt{5} - 11}{\left (2 \, \sqrt{5} + 5\right )} + 10 \, x\right ) + \sqrt{10} x \sqrt{5 \, \sqrt{5} - 11} \log \left (-\sqrt{10} \sqrt{5 \, \sqrt{5} - 11}{\left (2 \, \sqrt{5} + 5\right )} + 10 \, x\right ) - \sqrt{10} x \sqrt{5 \, \sqrt{5} + 11} \log \left (\sqrt{10} \sqrt{5 \, \sqrt{5} + 11}{\left (2 \, \sqrt{5} - 5\right )} + 10 \, x\right ) + \sqrt{10} x \sqrt{5 \, \sqrt{5} + 11} \log \left (-\sqrt{10} \sqrt{5 \, \sqrt{5} + 11}{\left (2 \, \sqrt{5} - 5\right )} + 10 \, x\right ) - 40}{40 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

1/40*(4*sqrt(10)*x*sqrt(5*sqrt(5) + 11)*arctan(1/40*(sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1)*(3*sqrt(5)*sqrt(2) - 5
*sqrt(2)) - 2*sqrt(10)*(3*sqrt(5)*x - 5*x))*sqrt(5*sqrt(5) + 11)) - 4*sqrt(10)*x*sqrt(5*sqrt(5) - 11)*arctan(1
/40*(sqrt(10)*sqrt(2*x^2 + sqrt(5) + 1)*(3*sqrt(5)*sqrt(2) + 5*sqrt(2)) - 2*sqrt(10)*(3*sqrt(5)*x + 5*x))*sqrt
(5*sqrt(5) - 11)) - sqrt(10)*x*sqrt(5*sqrt(5) - 11)*log(sqrt(10)*sqrt(5*sqrt(5) - 11)*(2*sqrt(5) + 5) + 10*x)
+ sqrt(10)*x*sqrt(5*sqrt(5) - 11)*log(-sqrt(10)*sqrt(5*sqrt(5) - 11)*(2*sqrt(5) + 5) + 10*x) - sqrt(10)*x*sqrt
(5*sqrt(5) + 11)*log(sqrt(10)*sqrt(5*sqrt(5) + 11)*(2*sqrt(5) - 5) + 10*x) + sqrt(10)*x*sqrt(5*sqrt(5) + 11)*l
og(-sqrt(10)*sqrt(5*sqrt(5) + 11)*(2*sqrt(5) - 5) + 10*x) - 40)/x

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Sympy [A]  time = 0.94756, size = 63, normalized size = 0.37 \begin{align*} \operatorname{RootSum}{\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{19251200 t^{7}}{11} - \frac{369792 t^{3}}{11} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{19251200 t^{7}}{11} - \frac{369792 t^{3}}{11} + x \right )} \right )\right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8-3*x**4+1),x)

[Out]

RootSum(6400*_t**4 - 880*_t**2 - 1, Lambda(_t, _t*log(19251200*_t**7/11 - 369792*_t**3/11 + x))) + RootSum(640
0*_t**4 + 880*_t**2 - 1, Lambda(_t, _t*log(19251200*_t**7/11 - 369792*_t**3/11 + x))) - 1/x

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Giac [A]  time = 1.23243, size = 205, normalized size = 1.19 \begin{align*} \frac{1}{20} \, \sqrt{50 \, \sqrt{5} - 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{20} \, \sqrt{50 \, \sqrt{5} + 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

1/20*sqrt(50*sqrt(5) - 110)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/20*sqrt(50*sqrt(5) + 110)*arctan(x/sqrt(1/2*
sqrt(5) - 1/2)) - 1/40*sqrt(50*sqrt(5) - 110)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) - 1
10)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) + 110)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2)))
- 1/40*sqrt(50*sqrt(5) + 110)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/x