3.406 \(\int \frac{x^{11}}{2+3 x^4+x^8} \, dx\)

Optimal. Leaf size=26 \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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Rubi [A]  time = 0.0189562, antiderivative size = 26, 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} \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^11/(2 + 3*x^4 + x^8),x]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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

Mathematica [A]  time = 0.0046948, size = 26, normalized size = 1. \[ \frac{x^4}{4}+\frac{1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(2 + 3*x^4 + x^8),x]

[Out]

x^4/4 + Log[1 + x^4]/4 - Log[2 + x^4]

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Maple [A]  time = 0.006, size = 23, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}}-\ln \left ({x}^{4}+2 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^8+3*x^4+2),x)

[Out]

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

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Maxima [A]  time = 1.0044, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (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+2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6815, size = 58, normalized size = 2.23 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (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+2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.125721, size = 19, normalized size = 0.73 \begin{align*} \frac{x^{4}}{4} + \frac{\log{\left (x^{4} + 1 \right )}}{4} - \log{\left (x^{4} + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**8+3*x**4+2),x)

[Out]

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

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Giac [A]  time = 1.10572, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac{1}{4} \, \log \left (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+2),x, algorithm="giac")

[Out]

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