3.389 \(\int \frac{x^5}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0532177, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1359, 1130, 207} \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - 3*x^4 + x^8),x]

[Out]

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

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 1130

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

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

Mathematica [A]  time = 0.03512, size = 91, normalized size = 1.12 \[ \frac{1}{40} \left (\left (\sqrt{5}-5\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5+\sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )-\left (\sqrt{5}-5\right ) \log \left (2 x^2+\sqrt{5}-1\right )-\left (5+\sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 - 3*x^4 + x^8),x]

[Out]

((-5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*x^2] - (-5 + Sqrt[5])*Log[-1 + S
qrt[5] + 2*x^2] - (5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/40

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Maple [A]  time = 0.003, size = 62, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{8}}-{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) }+{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}-{\frac{\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^8-3*x^4+1),x)

[Out]

-1/8*ln(x^4+x^2-1)-1/20*5^(1/2)*arctanh(1/5*(2*x^2+1)*5^(1/2))+1/8*ln(x^4-x^2-1)-1/20*5^(1/2)*arctanh(1/5*(2*x
^2-1)*5^(1/2))

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Maxima [B]  time = 1.50073, size = 117, normalized size = 1.44 \begin{align*} \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

1/40*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) + 1/40*sqrt(5)*log((2*x^2 - sqrt(5) - 1)/(2*x^2
+ sqrt(5) - 1)) - 1/8*log(x^4 + x^2 - 1) + 1/8*log(x^4 - x^2 - 1)

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Fricas [B]  time = 1.77738, size = 277, normalized size = 3.42 \begin{align*} \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} + 2 \, x^{2} - \sqrt{5}{\left (2 \, x^{2} + 1\right )} + 3}{x^{4} + x^{2} - 1}\right ) + \frac{1}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - 2 \, x^{2} - \sqrt{5}{\left (2 \, x^{2} - 1\right )} + 3}{x^{4} - x^{2} - 1}\right ) - \frac{1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

1/40*sqrt(5)*log((2*x^4 + 2*x^2 - sqrt(5)*(2*x^2 + 1) + 3)/(x^4 + x^2 - 1)) + 1/40*sqrt(5)*log((2*x^4 - 2*x^2
- sqrt(5)*(2*x^2 - 1) + 3)/(x^4 - x^2 - 1)) - 1/8*log(x^4 + x^2 - 1) + 1/8*log(x^4 - x^2 - 1)

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Sympy [B]  time = 0.447933, size = 165, normalized size = 2.04 \begin{align*} \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{3}{2} - \frac{3 \sqrt{5}}{10} - 640 \left (- \frac{1}{8} - \frac{\sqrt{5}}{40}\right )^{3} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{3}{2} - 640 \left (- \frac{1}{8} + \frac{\sqrt{5}}{40}\right )^{3} + \frac{3 \sqrt{5}}{10} \right )} + \left (\frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{3 \sqrt{5}}{10} - 640 \left (\frac{1}{8} - \frac{\sqrt{5}}{40}\right )^{3} + \frac{3}{2} \right )} + \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \log{\left (x^{2} - 640 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right )^{3} + \frac{3 \sqrt{5}}{10} + \frac{3}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**8-3*x**4+1),x)

[Out]

(-1/8 - sqrt(5)/40)*log(x**2 - 3/2 - 3*sqrt(5)/10 - 640*(-1/8 - sqrt(5)/40)**3) + (-1/8 + sqrt(5)/40)*log(x**2
- 3/2 - 640*(-1/8 + sqrt(5)/40)**3 + 3*sqrt(5)/10) + (1/8 - sqrt(5)/40)*log(x**2 - 3*sqrt(5)/10 - 640*(1/8 -
sqrt(5)/40)**3 + 3/2) + (sqrt(5)/40 + 1/8)*log(x**2 - 640*(sqrt(5)/40 + 1/8)**3 + 3*sqrt(5)/10 + 3/2)

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Giac [B]  time = 1.16845, size = 124, normalized size = 1.53 \begin{align*} \frac{1}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{8} \, \log \left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{8} \, \log \left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

1/40*sqrt(5)*log(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) + 1/40*sqrt(5)*log(abs(2*x^2 - sqrt(5) - 1)/a
bs(2*x^2 + sqrt(5) - 1)) - 1/8*log(abs(x^4 + x^2 - 1)) + 1/8*log(abs(x^4 - x^2 - 1))