3.162 \(\int \frac{x^4}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=112 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4 \sqrt [3]{3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{2 \sqrt [3]{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - (3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)])/2 + Log[1 + x]/6 - Log[3^(1/
3) + x]/(2*3^(1/3)) - Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(4*3^(1/3))

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Rubi [A]  time = 0.0680056, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1374, 292, 31, 634, 617, 204, 628, 618} \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4 \sqrt [3]{3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{2 \sqrt [3]{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(3 + 4*x^3 + x^6),x]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - (3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)])/2 + Log[1 + x]/6 - Log[3^(1/
3) + x]/(2*3^(1/3)) - Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(4*3^(1/3))

Rule 1374

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

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

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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]

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]

Rubi steps

\begin{align*} \int \frac{x^4}{3+4 x^3+x^6} \, dx &=-\left (\frac{1}{2} \int \frac{x}{1+x^3} \, dx\right )+\frac{3}{2} \int \frac{x}{3+x^3} \, dx\\ &=\frac{1}{6} \int \frac{1}{1+x} \, dx-\frac{1}{6} \int \frac{1+x}{1-x+x^2} \, dx-\frac{\int \frac{1}{\sqrt [3]{3}+x} \, dx}{2 \sqrt [3]{3}}+\frac{\int \frac{\sqrt [3]{3}+x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{2 \sqrt [3]{3}}\\ &=\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{2 \sqrt [3]{3}}-\frac{1}{12} \int \frac{-1+2 x}{1-x+x^2} \, dx-\frac{1}{4} \int \frac{1}{1-x+x^2} \, dx+\frac{3}{4} \int \frac{1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx+\frac{\int \frac{-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{4 \sqrt [3]{3}}\\ &=\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{2 \sqrt [3]{3}}-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4 \sqrt [3]{3}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{2} 3^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{\sqrt [3]{3}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac{1}{6} \log (1+x)-\frac{\log \left (\sqrt [3]{3}+x\right )}{2 \sqrt [3]{3}}-\frac{1}{12} \log \left (1-x+x^2\right )+\frac{\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4 \sqrt [3]{3}}\\ \end{align*}

Mathematica [A]  time = 0.0377344, size = 107, normalized size = 0.96 \[ \frac{1}{12} \left (-\log \left (x^2-x+1\right )+3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+2 \log (x+1)-2\ 3^{2/3} \log \left (3^{2/3} x+3\right )-6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(3 + 4*x^3 + x^6),x]

[Out]

(-6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - 2*3^(2/3)*
Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

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Maple [A]  time = 0.007, size = 84, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{6}}+{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{12}}+{\frac{\sqrt [6]{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^6+4*x^3+3),x)

[Out]

-1/12*ln(x^2-x+1)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))-1/6*3^(2/3)*ln(3^(1/3)+x)+1/12*3^(2/3)*ln(3^(2/3)-3^
(1/3)*x+x^2)+1/2*3^(1/6)*arctan(1/3*3^(1/2)*(2/3*3^(2/3)*x-1))+1/6*ln(1+x)

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Maxima [A]  time = 1.62697, size = 113, normalized size = 1.01 \begin{align*} \frac{1}{12} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/12*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/6*3^(2/3)*log(x + 3^(1/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2
*x - 1)) + 1/2*3^(1/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

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Fricas [A]  time = 1.51484, size = 392, normalized size = 3.5 \begin{align*} -\frac{1}{12} \cdot 3^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-3^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} x + x^{2} - 3^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}\right ) + \frac{1}{6} \cdot 3^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (3^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} + x\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \cdot 3^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, \left (-1\right )^{\frac{1}{3}} x + 3^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

-1/12*3^(2/3)*(-1)^(1/3)*log(-3^(1/3)*(-1)^(2/3)*x + x^2 - 3^(2/3)*(-1)^(1/3)) + 1/6*3^(2/3)*(-1)^(1/3)*log(3^
(1/3)*(-1)^(2/3) + x) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*3^(1/6)*(-1)^(1/3)*arctan(1/3*3^(1/6)*
(2*(-1)^(1/3)*x + 3^(1/3))) - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

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Sympy [C]  time = 0.619713, size = 134, normalized size = 1.2 \begin{align*} \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{2592 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5}}{5} + \frac{168 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2}}{5} \right )} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{168 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2}}{5} + \frac{2592 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5}}{5} \right )} + \operatorname{RootSum}{\left (24 t^{3} + 1, \left ( t \mapsto t \log{\left (\frac{2592 t^{5}}{5} + \frac{168 t^{2}}{5} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x + 2592*(-1/12 - sqrt(3)*I/12)**5/5 + 168*(-1/12 - sqrt(3)*I/12)**2
/5) + (-1/12 + sqrt(3)*I/12)*log(x + 168*(-1/12 + sqrt(3)*I/12)**2/5 + 2592*(-1/12 + sqrt(3)*I/12)**5/5) + Roo
tSum(24*_t**3 + 1, Lambda(_t, _t*log(2592*_t**5/5 + 168*_t**2/5 + x)))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError