3.4.87 \(\int \frac {x^9}{1-3 x^4+x^8} \, dx\) [387]

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]

1/2*x^2+1/2*arctanh(x^2*(1/2+1/2*5^(1/2)))*(1-2/5*5^(1/2))-1/2*arctanh(x^2*2^(1/2)/(3+5^(1/2))^(1/2))*(1+2/5*5
^(1/2))

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Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1136, 1180, 213} \begin {gather*} \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 ) \end {gather*}

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 213

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

Rule 1136

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 1180

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 1373

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]

Rubi steps

\begin {align*} \int \frac {x^9}{1-3 x^4+x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{1-3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{2} \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.04, size = 103, normalized size = 1.14 \begin {gather*} \frac {1}{20} \left (10 x^2+\left (-5+2 \sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x^2\right )+\left (5+2 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 x^2\right )+\left (5-2 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 x^2\right )-\left (5+2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x^2\right )\right ) \end {gather*}

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 = 0.04, size = 67, normalized size = 0.74

method	result	size
default	\(\frac {x^{2}}{2}+\frac {\ln \left (x^{4}-x^{2}-1\right )}{4}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{5}-\frac {\ln \left (x^{4}+x^{2}-1\right )}{4}-\frac {\arctanh \left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}\)	\(67\)
risch	\(\frac {x^{2}}{2}+\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right )}{4}+\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right ) \sqrt {5}}{10}+\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right )}{4}-\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right ) \sqrt {5}}{10}-\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right )}{4}+\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right ) \sqrt {5}}{10}-\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right )}{4}-\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right ) \sqrt {5}}{10}\)	\(131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*arctanh(1/5*(2*x^2+
1)*5^(1/2))*5^(1/2)

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Maxima [A]
time = 0.95, size = 92, normalized size = 1.02 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (50) = 100\).
time = 0.35, size = 114, normalized size = 1.27 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (63) = 126\).
time = 0.25, size = 170, normalized size = 1.89 \begin {gather*} \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 {gather*}

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 = 3.33, size = 97, normalized size = 1.08 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 1.33, size = 90, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}-\mathrm {atanh}\left (\frac {64\,x^2}{64\,\sqrt {5}+192}+\frac {64\,\sqrt {5}\,x^2}{64\,\sqrt {5}+192}\right )\,\left (\frac {\sqrt {5}}{5}+\frac {1}{2}\right )-\mathrm {atanh}\left (\frac {64\,x^2}{64\,\sqrt {5}-192}-\frac {64\,\sqrt {5}\,x^2}{64\,\sqrt {5}-192}\right )\,\left (\frac {\sqrt {5}}{5}-\frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^8 - 3*x^4 + 1),x)

[Out]

x^2/2 - atanh((64*x^2)/(64*5^(1/2) + 192) + (64*5^(1/2)*x^2)/(64*5^(1/2) + 192))*(5^(1/2)/5 + 1/2) - atanh((64
*x^2)/(64*5^(1/2) - 192) - (64*5^(1/2)*x^2)/(64*5^(1/2) - 192))*(5^(1/2)/5 - 1/2)

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