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

Optimal. Leaf size=89 \[ -\frac{1}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

[Out]

-1/(2*x^2) - (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + ((3 + Sqrt[5])^(3/2)*ArcTanh[Sqr
t[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

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Rubi [A]  time = 0.0576259, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1359, 1123, 1166, 207} \[ -\frac{1}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(2*x^2) - (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + ((3 + Sqrt[5])^(3/2)*ArcTanh[Sqr
t[(3 + Sqrt[5])/2]*x^2])/(4*Sqrt[10])

Rule 1359

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[
1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k) + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b,
c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 1123

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2 +
c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (1-3 x^4+x^8\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{2 x^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{3-x^2}{1-3 x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{2 x^2}+\frac{1}{20} \left (-5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )-\frac{1}{20} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{2 x^2}-\frac{1}{10} \sqrt{45-20 \sqrt{5}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{4 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0602255, size = 103, normalized size = 1.16 \[ \frac{1}{20} \left (-\frac{10}{x^2}-\left (5+2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5-2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+\left (5+2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )+\left (2 \sqrt{5}-5\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10/x^2 - (5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 - 2*Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] + (5 + 2*Sqrt[
5])*Log[-1 + Sqrt[5] + 2*x^2] + (-5 + 2*Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/20

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Maple [A]  time = 0.009, size = 67, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{4}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{4}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{1}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x^4-x^2-1)+1/5*5^(1/2)*arctanh(1/5*(2*x^2-1)*5^(1/2))-1/4*ln(x^4+x^2-1)+1/5*5^(1/2)*arctanh(1/5*(2*x^2+
1)*5^(1/2))-1/2/x^2

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Maxima [A]  time = 1.47979, size = 124, normalized size = 1.39 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{4} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{4} \, \log \left (x^{4} - x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/10*sqrt(5)*log((2*x^2 - sqrt(5) - 1)/(2*x^2
+ sqrt(5) - 1)) - 1/2/x^2 - 1/4*log(x^4 + x^2 - 1) + 1/4*log(x^4 - x^2 - 1)

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Fricas [B]  time = 1.75756, size = 306, normalized size = 3.44 \begin{align*} \frac{2 \, \sqrt{5} x^{2} \log \left (\frac{2 \, x^{4} + 2 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} + 1\right )} + 3}{x^{4} + x^{2} - 1}\right ) + 2 \, \sqrt{5} x^{2} \log \left (\frac{2 \, x^{4} - 2 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} - 1\right )} + 3}{x^{4} - x^{2} - 1}\right ) - 5 \, x^{2} \log \left (x^{4} + x^{2} - 1\right ) + 5 \, x^{2} \log \left (x^{4} - x^{2} - 1\right ) - 10}{20 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(2*sqrt(5)*x^2*log((2*x^4 + 2*x^2 + sqrt(5)*(2*x^2 + 1) + 3)/(x^4 + x^2 - 1)) + 2*sqrt(5)*x^2*log((2*x^4
- 2*x^2 + sqrt(5)*(2*x^2 - 1) + 3)/(x^4 - x^2 - 1)) - 5*x^2*log(x^4 + x^2 - 1) + 5*x^2*log(x^4 - x^2 - 1) - 10
)/x^2

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Sympy [B]  time = 0.476674, size = 172, normalized size = 1.93 \begin{align*} \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - \frac{123}{8} - \frac{123 \sqrt{5}}{20} + 280 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} \right )} + \left (\frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{123}{8} + 280 \left (\frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} + \frac{123 \sqrt{5}}{20} \right )} + \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{123 \sqrt{5}}{20} + 280 \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right )^{3} + \frac{123}{8} \right )} + \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} + 280 \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} + \frac{123 \sqrt{5}}{20} + \frac{123}{8} \right )} - \frac{1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(5)/10 + 1/4)*log(x**2 - 123/8 - 123*sqrt(5)/20 + 280*(sqrt(5)/10 + 1/4)**3) + (1/4 - sqrt(5)/10)*log(x**
2 - 123/8 + 280*(1/4 - sqrt(5)/10)**3 + 123*sqrt(5)/20) + (-1/4 + sqrt(5)/10)*log(x**2 - 123*sqrt(5)/20 + 280*
(-1/4 + sqrt(5)/10)**3 + 123/8) + (-1/4 - sqrt(5)/10)*log(x**2 + 280*(-1/4 - sqrt(5)/10)**3 + 123*sqrt(5)/20 +
123/8) - 1/(2*x**2)

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Giac [A]  time = 1.16964, size = 131, normalized size = 1.47 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{4} \, \log \left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(5)*log(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 1/10*sqrt(5)*log(abs(2*x^2 - sqrt(5) - 1)/
abs(2*x^2 + sqrt(5) - 1)) - 1/2/x^2 - 1/4*log(abs(x^4 + x^2 - 1)) + 1/4*log(abs(x^4 - x^2 - 1))