3.387 \(\int \frac{x^9}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=90 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0720074, antiderivative size = 90, 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, 1122, 1166, 207} \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(1 - 3*x^4 + x^8),x]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt
[(3 + Sqrt[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 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[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{x^9}{1-3 x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{1-3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1-3 x^2}{1-3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{20} \left (-15+7 \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 (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{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{1}{20} \sqrt{180-80 \sqrt{5}} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0509876, size = 103, normalized size = 1.14 \[ \frac{1}{20} \left (10 x^2+\left (2 \sqrt{5}-5\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5+2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+\left (5-2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )-\left (5+2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(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 = 0.005, size = 67, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{2}}-{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^8-3*x^4+1),x)

[Out]

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

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Maxima [A]  time = 1.46521, size = 124, normalized size = 1.38 \begin{align*} \frac{1}{2} \, x^{2} + \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}{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(x^9/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.71002, size = 290, normalized size = 3.22 \begin{align*} \frac{1}{2} \, x^{2} + \frac{1}{10} \, \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}{10} \, \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}{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(x^9/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.462109, size = 170, normalized size = 1.89 \begin{align*} \frac{x^{2}}{2} + \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{47}{8} - \frac{47 \sqrt{5}}{20} - 120 \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} \right )} + \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{47}{8} - 120 \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right )^{3} + \frac{47 \sqrt{5}}{20} \right )} + \left (\frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{47 \sqrt{5}}{20} - 120 \left (\frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} + \frac{47}{8} \right )} + \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - 120 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} + \frac{47 \sqrt{5}}{20} + \frac{47}{8} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(x**8-3*x**4+1),x)

[Out]

x**2/2 + (-1/4 - sqrt(5)/10)*log(x**2 - 47/8 - 47*sqrt(5)/20 - 120*(-1/4 - sqrt(5)/10)**3) + (-1/4 + sqrt(5)/1
0)*log(x**2 - 47/8 - 120*(-1/4 + sqrt(5)/10)**3 + 47*sqrt(5)/20) + (1/4 - sqrt(5)/10)*log(x**2 - 47*sqrt(5)/20
- 120*(1/4 - sqrt(5)/10)**3 + 47/8) + (sqrt(5)/10 + 1/4)*log(x**2 - 120*(sqrt(5)/10 + 1/4)**3 + 47*sqrt(5)/20
+ 47/8)

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Giac [A]  time = 1.17847, size = 131, normalized size = 1.46 \begin{align*} \frac{1}{2} \, x^{2} + \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}{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(x^9/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

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