3.388 \(\int \frac{x^7}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]

[Out]

((5 - 3*Sqrt[5])*Log[3 - Sqrt[5] - 2*x^4])/40 + ((5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40

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Rubi [A]  time = 0.0318653, antiderivative size = 55, 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 = {1357, 632, 31} \[ \frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (-2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 - 3*x^4 + x^8),x]

[Out]

((5 - 3*Sqrt[5])*Log[3 - Sqrt[5] - 2*x^4])/40 + ((5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40

Rule 1357

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

Rule 632

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^7}{1-3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{1-3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{40} \left (5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )\\ &=\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (3-\sqrt{5}-2 x^4\right )+\frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (3+\sqrt{5}-2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0208405, size = 53, normalized size = 0.96 \[ \frac{1}{40} \left (5+3 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}+3\right )+\frac{1}{40} \left (5-3 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 - 3*x^4 + x^8),x]

[Out]

((5 + 3*Sqrt[5])*Log[3 + Sqrt[5] - 2*x^4])/40 + ((5 - 3*Sqrt[5])*Log[-3 + Sqrt[5] + 2*x^4])/40

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Maple [A]  time = 0.002, size = 33, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{8}-3\,{x}^{4}+1 \right ) }{8}}-{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}-3 \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^8-3*x^4+1),x)

[Out]

1/8*ln(x^8-3*x^4+1)-3/20*5^(1/2)*arctanh(1/5*(2*x^4-3)*5^(1/2))

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Maxima [A]  time = 1.47837, size = 61, normalized size = 1.11 \begin{align*} \frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} - 3}{2 \, x^{4} + \sqrt{5} - 3}\right ) + \frac{1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72496, size = 143, normalized size = 2.6 \begin{align*} \frac{3}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{8} - 6 \, x^{4} - \sqrt{5}{\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) + \frac{1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

3/40*sqrt(5)*log((2*x^8 - 6*x^4 - sqrt(5)*(2*x^4 - 3) + 7)/(x^8 - 3*x^4 + 1)) + 1/8*log(x^8 - 3*x^4 + 1)

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Sympy [A]  time = 0.137365, size = 53, normalized size = 0.96 \begin{align*} \left (\frac{1}{8} + \frac{3 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{3}{2} - \frac{\sqrt{5}}{2} \right )} + \left (\frac{1}{8} - \frac{3 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{3}{2} + \frac{\sqrt{5}}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**8-3*x**4+1),x)

[Out]

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

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Giac [A]  time = 1.14867, size = 65, normalized size = 1.18 \begin{align*} \frac{3}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{4} - \sqrt{5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt{5} - 3 \right |}}\right ) + \frac{1}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

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