\(\int \frac {1}{3+4 x^3+x^6} \, dx\) [165]

Optimal result

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

[Out]

1/18*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))+1/6*ln(1+x)-1/18*3^(1/3)*ln(3^(1/3)+x)-1/12*ln(x^2-x+1)+1/36*3^
(1/3)*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 = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1361, 206, 31, 648, 632, 210, 642, 631} \[ \int \frac {1}{3+4 x^3+x^6} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6 \sqrt [6]{3}}-\frac {1}{12} \log \left (x^2-x+1\right )+\frac {\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12\ 3^{2/3}}+\frac {1}{6} \log (x+1)-\frac {\log \left (x+\sqrt [3]{3}\right )}{6\ 3^{2/3}} \]

[In]

Int[(3 + 4*x^3 + x^6)^(-1),x]

[Out]

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

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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 1361

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{1+x^3} \, dx-\frac {1}{2} \int \frac {1}{3+x^3} \, dx \\ & = \frac {1}{6} \int \frac {1}{1+x} \, dx+\frac {1}{6} \int \frac {2-x}{1-x+x^2} \, dx-\frac {\int \frac {1}{\sqrt [3]{3}+x} \, dx}{6\ 3^{2/3}}-\frac {\int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{6\ 3^{2/3}} \\ & = \frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{6\ 3^{2/3}}-\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 {\int \frac {-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx}{12\ 3^{2/3}}-\frac {\int \frac {1}{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 )}{6\ 3^{2/3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{12\ 3^{2/3}}-\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\ 3^{2/3}} \\ & = -\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 )}{6 \sqrt [6]{3}}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{6\ 3^{2/3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{12\ 3^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=\frac {1}{36} \left (2\ 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+6 \log (1+x)-2 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )-3 \log \left (1-x+x^2\right )+\sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]

[In]

Integrate[(3 + 4*x^3 + x^6)^(-1),x]

[Out]

(2*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 6*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 6*Log[1 + x] - 2*3^(1/3)*L
og[3 + 3^(2/3)*x] - 3*Log[1 - x + x^2] + 3^(1/3)*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 = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (x -3 \textit {\_R} \right )\right )}{6}-\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}\)	\(58\)
default	\(\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{18}+\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{36}-\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{18}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\)	\(84\)

[In]

int(1/(x^6+4*x^3+3),x,method=_RETURNVERBOSE)

[Out]

1/6*sum(_R*ln(x-3*_R),_R=RootOf(9*_Z^3+1))-1/12*ln(4*x^2-4*x+4)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/6*ln
(x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.11 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=\frac {1}{18} \cdot 9^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{108} \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x + 3 \, x^{2} + 3 \cdot 9^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right ) + \frac {1}{54} \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, x\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]

[In]

integrate(1/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/18*9^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/27*9^(1/6)*(2*9^(2/3)*sqrt(3)*(-1)^(2/3)*x - 3*9^(1/3)*sqrt(3))) - 1/
108*9^(2/3)*(-1)^(1/3)*log(9^(2/3)*(-1)^(1/3)*x + 3*x^2 + 3*9^(1/3)*(-1)^(2/3)) + 1/54*9^(2/3)*(-1)^(1/3)*log(
-9^(2/3)*(-1)^(1/3) + 3*x) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 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 = 1.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.11 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=\frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {13}{10} - \frac {13 \sqrt {3} i}{10} + \frac {23328 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{5} \right )} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {13}{10} + \frac {23328 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{5} + \frac {13 \sqrt {3} i}{10} \right )} + \operatorname {RootSum} {\left (1944 t^{3} + 1, \left ( t \mapsto t \log {\left (\frac {23328 t^{4}}{5} - \frac {78 t}{5} + x \right )} \right )\right )} \]

[In]

integrate(1/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 + (-1/12 + sqrt(3)*I/12)*log(x + 13/10 - 13*sqrt(3)*I/10 + 23328*(-1/12 + sqrt(3)*I/12)**4/5) + (
-1/12 - sqrt(3)*I/12)*log(x + 13/10 + 23328*(-1/12 - sqrt(3)*I/12)**4/5 + 13*sqrt(3)*I/10) + RootSum(1944*_t**
3 + 1, Lambda(_t, _t*log(23328*_t**4/5 - 78*_t/5 + x)))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=-\frac {1}{18} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{36} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]

[In]

integrate(1/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

-1/18*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/36*3^(1/3)*l
og(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/18*3^(1/3)*log(x + 3^(1/3)) - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=-\frac {1}{18} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{36} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

integrate(1/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

-1/18*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/36*3^(1/3)*l
og(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/18*3^(1/3)*log(abs(x + 3^(1/3))) - 1/12*log(x^2 - x + 1) + 1/6*log(abs(x + 1
))

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {1}{3+4 x^3+x^6} \, dx=\frac {\ln \left (x+1\right )}{6}-\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{18}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}\,\ln \left (x-{\left (-1\right )}^{1/3}\,3^{1/3}\right )}{18}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}+\frac {{\left (-1\right )}^{1/3}\,3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \]

[In]

int(1/(4*x^3 + x^6 + 3),x)

[Out]

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