\(\int \frac {x}{3+4 x^3+x^6} \, dx\) [164]

Optimal result

Integrand size = 14, antiderivative size = 112 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2\ 3^{5/6}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{6 \sqrt [3]{3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{12 \sqrt [3]{3}} \]

[Out]

1/6*3^(1/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/18*3^(2/3)*ln(3^(1/3)+x)+1/12*ln(x^2-x+1)-1/36*3^(
2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1389, 298, 31, 648, 632, 210, 642, 631} \[ \int \frac {x}{3+4 x^3+x^6} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2\ 3^{5/6}}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12 \sqrt [3]{3}}-\frac {1}{6} \log (x+1)+\frac {\log \left (x+\sqrt [3]{3}\right )}{6 \sqrt [3]{3}} \]

[In]

Int[x/(3 + 4*x^3 + x^6),x]

[Out]

-1/2*ArcTan[(1 - 2*x)/Sqrt[3]]/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 31

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

Rule 210

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

Rule 298

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 631

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 632

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 642

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 648

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 1389

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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {\text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=\frac {1}{36} \left (6 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-6 \log (1+x)+2\ 3^{2/3} \log \left (3+3^{2/3} x\right )+3 \log \left (1-x+x^2\right )-3^{2/3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.50

method	result	size
risch	\(-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+x \right )\right )}{6}\)	\(56\)
default	\(-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{18}-\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{36}-\frac {3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\)	\(84\)

[In]

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

[Out]

-1/6*ln(x+1)+1/12*ln(x^2-x+1)+1/6*3^(1/2)*arctan(2/3*(x-1/2)*3^(1/2))+1/6*sum(_R*ln(3*_R^2+x),_R=RootOf(3*_Z^3
-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=-\frac {1}{36} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{6}} \arctan \left (-\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=- \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + 90 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2} + 11664 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + 11664 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5} + 90 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2} \right )} + \operatorname {RootSum} {\left (648 t^{3} - 1, \left ( t \mapsto t \log {\left (11664 t^{5} + 90 t^{2} + x \right )} \right )\right )} \]

[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)))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=-\frac {1}{36} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=-\frac {1}{36} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

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

[Out]

-1/36*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/18*3^(2/3)*log(abs(x + 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqr
t(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(abs(x + 1)
)

Mupad [B] (verification not implemented)

Time = 8.48 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01 \[ \int \frac {x}{3+4 x^3+x^6} \, dx=\frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{18}-\frac {\ln \left (x+1\right )}{6}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{36}-\frac {3^{1/6}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{36}+\frac {3^{1/6}\,1{}\mathrm {i}}{12}\right ) \]

[In]

int(x/(4*x^3 + x^6 + 3),x)

[Out]

(3^(2/3)*log(x + 3^(1/3)))/18 - log(x + 1)/6 - log(x - (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/12 - 1/12) + log(x
+ (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/12 + 1/12) - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(2/3)/36 - (3^(1/6)*
1i)/12) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(2/3)/36 + (3^(1/6)*1i)/12)