\(\int \frac {x^9}{3+4 x^3+x^6} \, dx\) [159]

Optimal result

Integrand size = 16, antiderivative size = 122 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=-4 x+\frac {x^4}{4}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {3}{2} 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-\frac {1}{6} \log (1+x)+\frac {3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {3}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right ) \]

[Out]

-4*x+1/4*x^4-3/2*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+3/2*3^(1/3)*ln(3^(1/3)+x)+1/12*ln(x^2-x
+1)-3/4*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 = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1381, 1516, 1436, 206, 31, 648, 632, 210, 642, 631} \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {3}{2} 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac {x^4}{4}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {3}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-4 x-\frac {1}{6} \log (x+1)+\frac {3}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right ) \]

[In]

Int[x^9/(3 + 4*x^3 + x^6),x]

[Out]

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

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 1381

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[d^(2*n - 1)*(d*x)^
(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + 2*n*p + 1))), x] - Dist[d^(2*n)/(c*(m + 2*n*p + 1)), In
t[(d*x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x^n + c*x^(2*n))^p, x], x] /; Fr
eeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n
*p + 1, 0] && IntegerQ[p]

Rule 1436

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

Rule 1516

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Dist[f^n
/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p +
1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {x^4}{4}-\frac {1}{4} \int \frac {x^3 \left (12+16 x^3\right )}{3+4 x^3+x^6} \, dx \\ & = -4 x+\frac {x^4}{4}+\frac {1}{4} \int \frac {48+52 x^3}{3+4 x^3+x^6} \, dx \\ & = -4 x+\frac {x^4}{4}-\frac {1}{2} \int \frac {1}{1+x^3} \, dx+\frac {27}{2} \int \frac {1}{3+x^3} \, dx \\ & = -4 x+\frac {x^4}{4}-\frac {1}{6} \int \frac {1}{1+x} \, dx-\frac {1}{6} \int \frac {2-x}{1-x+x^2} \, dx+\frac {1}{2} \left (3 \sqrt [3]{3}\right ) \int \frac {1}{\sqrt [3]{3}+x} \, dx+\frac {1}{2} \left (3 \sqrt [3]{3}\right ) \int \frac {2 \sqrt [3]{3}-x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx \\ & = -4 x+\frac {x^4}{4}-\frac {1}{6} \log (1+x)+\frac {3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\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} \left (3 \sqrt [3]{3}\right ) \int \frac {-\sqrt [3]{3}+2 x}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx+\frac {1}{4} \left (9\ 3^{2/3}\right ) \int \frac {1}{3^{2/3}-\sqrt [3]{3} x+x^2} \, dx \\ & = -4 x+\frac {x^4}{4}-\frac {1}{6} \log (1+x)+\frac {3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {3}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} \left (9 \sqrt [3]{3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{3}}\right ) \\ & = -4 x+\frac {x^4}{4}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {3}{2} 3^{5/6} \tan ^{-1}\left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-\frac {1}{6} \log (1+x)+\frac {3}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {3}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {1}{12} \left (-48 x+3 x^4-18\ 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \log (1+x)+18 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )+\log \left (1-x+x^2\right )-9 \sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]

[In]

Integrate[x^9/(3 + 4*x^3 + x^6),x]

[Out]

(-48*x + 3*x^4 - 18*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 +
x] + 18*3^(1/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 9*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

Maple [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.51

method	result	size
risch	\(\frac {x^{4}}{4}-4 x +\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-3\right )}{\sum }\textit {\_R} \ln \left (x +\textit {\_R} \right )\right )}{2}\)	\(62\)
default	\(\frac {x^{4}}{4}-4 x -\frac {\ln \left (x +1\right )}{6}+\frac {3 \,3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{2}-\frac {3 \,3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{4}+\frac {3 \,3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\)	\(92\)

[In]

int(x^9/(x^6+4*x^3+3),x,method=_RETURNVERBOSE)

[Out]

1/4*x^4-4*x+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)+3/2*sum(_R*ln(x+_R),_R=Ro
otOf(_Z^3-3))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {1}{4} \, x^{4} + \frac {3}{2} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {2}{3} \cdot 3^{\frac {1}{6}} x - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{4} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {3}{2} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - 4 \, x + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

[In]

integrate(x^9/(x^6+4*x^3+3),x, algorithm="fricas")

[Out]

1/4*x^4 + 3/2*3^(5/6)*arctan(2/3*3^(1/6)*x - 1/3*sqrt(3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 3/4*3^
(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 3/2*3^(1/3)*log(x + 3^(1/3)) - 4*x + 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 = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.06 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {x^{4}}{4} - 4 x - \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {9841}{19692} - \frac {9841 \sqrt {3} i}{19692} + \frac {360 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{547} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {9841}{19692} + \frac {360 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{547} + \frac {9841 \sqrt {3} i}{19692} \right )} + \operatorname {RootSum} {\left (8 t^{3} - 81, \left ( t \mapsto t \log {\left (\frac {360 t^{4}}{547} - \frac {9841 t}{1641} + x \right )} \right )\right )} \]

[In]

integrate(x**9/(x**6+4*x**3+3),x)

[Out]

x**4/4 - 4*x - log(x + 1)/6 + (1/12 + sqrt(3)*I/12)*log(x - 9841/19692 - 9841*sqrt(3)*I/19692 + 360*(1/12 + sq
rt(3)*I/12)**4/547) + (1/12 - sqrt(3)*I/12)*log(x - 9841/19692 + 360*(1/12 - sqrt(3)*I/12)**4/547 + 9841*sqrt(
3)*I/19692) + RootSum(8*_t**3 - 81, Lambda(_t, _t*log(360*_t**4/547 - 9841*_t/1641 + x)))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.75 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {1}{4} \, x^{4} + \frac {3}{2} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{4} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {3}{2} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - 4 \, x + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]

[In]

integrate(x^9/(x^6+4*x^3+3),x, algorithm="maxima")

[Out]

1/4*x^4 + 3/2*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)) - 3/4*3^
(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 3/2*3^(1/3)*log(x + 3^(1/3)) - 4*x + 1/12*log(x^2 - x + 1) - 1/6*log(x
+ 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.77 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {1}{4} \, x^{4} + \frac {3}{2} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{4} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {3}{2} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - 4 \, x + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

integrate(x^9/(x^6+4*x^3+3),x, algorithm="giac")

[Out]

1/4*x^4 + 3/2*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)) - 3/4*3^
(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 3/2*3^(1/3)*log(abs(x + 3^(1/3))) - 4*x + 1/12*log(x^2 - x + 1) - 1/6*l
og(abs(x + 1))

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {x^9}{3+4 x^3+x^6} \, dx=\frac {3\,3^{1/3}\,\ln \left (x+3^{1/3}\right )}{2}-\frac {\ln \left (x+1\right )}{6}-4\,x+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {x^4}{4}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3\,3^{1/3}}{4}+\frac {3^{5/6}\,3{}\mathrm {i}}{4}\right )+3^{1/3}\,\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right ) \]

[In]

int(x^9/(4*x^3 + x^6 + 3),x)

[Out]

(3*3^(1/3)*log(x + 3^(1/3)))/2 - log(x + 1)/6 - 4*x + 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) + x^4/4 - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*((3*3^(1
/3))/4 + (3^(5/6)*3i)/4) + 3^(1/3)*log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*((3^(1/2)*3i)/4 - 3/4)