3.368 \(\int \frac{x^{11}}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=62 \[ \frac{x^4}{4}-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right ) \]

[Out]

x^4/4 - ((15 - 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 - ((15 + 7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Rubi [A]  time = 0.0562573, antiderivative size = 62, 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 = {1357, 703, 632, 31} \[ \frac{x^4}{4}-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^11/(1 + 3*x^4 + x^8),x]

[Out]

x^4/4 - ((15 - 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 - ((15 + 7*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 703

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

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^{11}}{1+3 x^4+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-1-3 x}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}+\frac{1}{40} \left (-15+7 \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 (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )\\ &=\frac{x^4}{4}-\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (3-\sqrt{5}+2 x^4\right )-\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (3+\sqrt{5}+2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0321734, size = 57, normalized size = 0.92 \[ \frac{1}{40} \left (10 x^4+\left (7 \sqrt{5}-15\right ) \log \left (-2 x^4+\sqrt{5}-3\right )-\left (15+7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(1 + 3*x^4 + x^8),x]

[Out]

(10*x^4 + (-15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] - (15 + 7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4])/40

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Maple [A]  time = 0.003, size = 38, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}}{4}}-{\frac{3\,\ln \left ({x}^{8}+3\,{x}^{4}+1 \right ) }{8}}-{\frac{7\,\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^11/(x^8+3*x^4+1),x)

[Out]

1/4*x^4-3/8*ln(x^8+3*x^4+1)-7/20*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)

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Maxima [A]  time = 1.49733, size = 68, normalized size = 1.1 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{3}{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^11/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48317, size = 157, normalized size = 2.53 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{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{3}{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^11/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.142039, size = 60, normalized size = 0.97 \begin{align*} \frac{x^{4}}{4} + \left (- \frac{3}{8} + \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (- \frac{7 \sqrt{5}}{40} - \frac{3}{8}\right ) \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**8+3*x**4+1),x)

[Out]

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

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Giac [A]  time = 1.22097, size = 68, normalized size = 1.1 \begin{align*} \frac{1}{4} \, x^{4} + \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{3}{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^11/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

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