3.4.89 \(\int \frac {x^5}{1-3 x^4+x^8} \, dx\) [389]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 = {1373, 1144, 213} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - 3*x^4 + x^8),x]

[Out]

-1/2*(Sqrt[(3 + Sqrt[5])/10]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2]) + (Sqrt[(3 - Sqrt[5])/10]*ArcTanh[Sqrt[(3 + S
qrt[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 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), 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 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^5}{1-3 x^4+x^8} \, dx &=\frac {1}{2} \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.02, size = 91, normalized size = 1.12 \begin {gather*} \frac {1}{40} \left (\left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x^2\right )+\left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 x^2\right )-\left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 x^2\right )-\left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x^2\right )\right ) \end {gather*}

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 = 0.03, size = 62, normalized size = 0.77

method	result	size
default	\(\frac {\ln \left (x^{4}-x^{2}-1\right )}{8}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{20}-\frac {\ln \left (x^{4}+x^{2}-1\right )}{8}-\frac {\arctanh \left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{20}\)	\(62\)
risch	\(\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right )}{8}+\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right ) \sqrt {5}}{40}+\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right )}{8}-\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right ) \sqrt {5}}{40}-\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right )}{8}+\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right ) \sqrt {5}}{40}-\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right )}{8}-\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right ) \sqrt {5}}{40}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (41) = 82\).
time = 0.51, size = 87, normalized size = 1.07 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (41) = 82\).
time = 0.39, size = 109, normalized size = 1.35 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (58) = 116\).
time = 0.17, size = 165, normalized size = 2.04 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (41) = 82\).
time = 3.59, size = 92, normalized size = 1.14 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 1.38, size = 77, normalized size = 0.95 \begin {gather*} -\mathrm {atanh}\left (\frac {4\,x^2}{\sqrt {5}-3}-\frac {2\,\sqrt {5}\,x^2}{\sqrt {5}-3}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{4}\right )-\mathrm {atanh}\left (\frac {4\,x^2}{\sqrt {5}+3}+\frac {2\,\sqrt {5}\,x^2}{\sqrt {5}+3}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^8 - 3*x^4 + 1),x)

[Out]

- atanh((4*x^2)/(5^(1/2) - 3) - (2*5^(1/2)*x^2)/(5^(1/2) - 3))*(5^(1/2)/20 + 1/4) - atanh((4*x^2)/(5^(1/2) + 3
) + (2*5^(1/2)*x^2)/(5^(1/2) + 3))*(5^(1/2)/20 - 1/4)

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