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

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

[Out]

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

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Rubi [A]  time = 0.0690096, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {1375, 292, 31, 634, 618, 204, 628, 617} \[ \frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12 \sqrt [3]{3}}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{6 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2\ 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[x/(3 + 4*x^3 + x^6),x]

[Out]

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

Rule 1375

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

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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0378439, size = 108, normalized size = 0.96 \[ \frac{1}{36} \left (3 \log \left (x^2-x+1\right )-3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-6 \log (x+1)+2\ 3^{2/3} \log \left (3^{2/3} x+3\right )+6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(3 + 4*x^3 + x^6),x]

[Out]

(6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 6*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 6*Log[1 + x] + 2*3^(2/3)*L
og[3 + 3^(2/3)*x] + 3*Log[1 - x + x^2] - 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/36

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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 ) }{18}}-{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{36}}-{\frac{\sqrt [6]{3}}{6}\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/(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))+1/18*3^(2/3)*ln(3^(1/3)+x)-1/36*3^(2/3)*ln(3^(2/3)-3^
(1/3)*x+x^2)-1/6*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 = 1.65587, size = 113, normalized size = 1.01 \begin{align*} -\frac{1}{36} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{18} \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}{6} \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/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

-1/36*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/18*3^(2/3)*log(x + 3^(1/3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*
(2*x - 1)) - 1/6*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 = 1.49911, size = 289, normalized size = 2.58 \begin{align*} -\frac{1}{36} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{18} \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}{6} \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/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

-1/36*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/18*3^(2/3)*log(x + 3^(1/3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*
(2*x - 1)) + 1/6*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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Sympy [C]  time = 1.30997, size = 119, normalized size = 1.06 \begin{align*} - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + 90 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2} + 11664 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5} \right )} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + 11664 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5} + 90 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2} \right )} + \operatorname{RootSum}{\left (648 t^{3} - 1, \left ( t \mapsto t \log{\left (11664 t^{5} + 90 t^{2} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 90*(1/12 - sqrt(3)*I/12)**2 + 11664*(1/12 - sqrt(3)*I/12)**5) +
(1/12 + sqrt(3)*I/12)*log(x + 11664*(1/12 + sqrt(3)*I/12)**5 + 90*(1/12 + sqrt(3)*I/12)**2) + RootSum(648*_t**
3 - 1, Lambda(_t, _t*log(11664*_t**5 + 90*_t**2 + 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/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError