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

Optimal. Leaf size=37 \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

[Out]

-ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[1 + x^4 + x^8]/8

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Rubi [A]  time = 0.0310579, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1357, 634, 618, 204, 628} \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 + x^4 + x^8),x]

[Out]

-ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[1 + x^4 + x^8]/8

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 618

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

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.008011, size = 37, normalized size = 1. \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 + x^4 + x^8),x]

[Out]

-ArcTan[(1 + 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[1 + x^4 + x^8]/8

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Maple [A]  time = 0.003, size = 31, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{8}+{x}^{4}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}+1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*ln(x^8+x^4+1)-1/12*arctan(1/3*(2*x^4+1)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.49953, size = 41, normalized size = 1.11 \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) + 1/8*log(x^8 + x^4 + 1)

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Fricas [A]  time = 1.48928, size = 97, normalized size = 2.62 \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) + 1/8*log(x^8 + x^4 + 1)

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Sympy [A]  time = 0.130603, size = 37, normalized size = 1. \begin{align*} \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**8 + x**4 + 1)/8 - sqrt(3)*atan(2*sqrt(3)*x**4/3 + sqrt(3)/3)/12

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Giac [A]  time = 1.09363, size = 41, normalized size = 1.11 \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 + 1)) + 1/8*log(x^8 + x^4 + 1)