\(\int \frac {x^2}{13-6 x^3+x^6} \, dx\) [45]

Optimal result

Integrand size = 16, antiderivative size = 14 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \arctan \left (\frac {1}{2} \left (-3+x^3\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 632, 210} \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \arctan \left (\frac {1}{2} \left (x^3-3\right )\right ) \]

[In]

Int[x^2/(13 - 6*x^3 + x^6),x]

[Out]

ArcTan[(-3 + x^3)/2]/6

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 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 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{13-6 x+x^2} \, dx,x,x^3\right ) \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2 \left (-3+x^3\right )\right )\right ) \\ & = \frac {1}{6} \arctan \left (\frac {1}{2} \left (-3+x^3\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \arctan \left (\frac {1}{2} \left (-3+x^3\right )\right ) \]

[In]

Integrate[x^2/(13 - 6*x^3 + x^6),x]

[Out]

ArcTan[(-3 + x^3)/2]/6

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\arctan \left (\frac {x^{3}}{2}-\frac {3}{2}\right )}{6}\)	\(11\)
risch	\(\frac {\arctan \left (\frac {x^{3}}{2}-\frac {3}{2}\right )}{6}\)	\(11\)
parallelrisch	\(\frac {i \ln \left (x^{3}+2 i-3\right )}{12}-\frac {i \ln \left (x^{3}-2 i-3\right )}{12}\)	\(24\)

[In]

int(x^2/(x^6-6*x^3+13),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \]

[In]

integrate(x^2/(x^6-6*x^3+13),x, algorithm="fricas")

[Out]

1/6*arctan(1/2*x^3 - 3/2)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {\operatorname {atan}{\left (\frac {x^{3}}{2} - \frac {3}{2} \right )}}{6} \]

[In]

integrate(x**2/(x**6-6*x**3+13),x)

[Out]

atan(x**3/2 - 3/2)/6

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \]

[In]

integrate(x^2/(x^6-6*x^3+13),x, algorithm="maxima")

[Out]

1/6*arctan(1/2*x^3 - 3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {1}{6} \, \arctan \left (\frac {1}{2} \, x^{3} - \frac {3}{2}\right ) \]

[In]

integrate(x^2/(x^6-6*x^3+13),x, algorithm="giac")

[Out]

1/6*arctan(1/2*x^3 - 3/2)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{13-6 x^3+x^6} \, dx=\frac {\mathrm {atan}\left (\frac {x^3}{2}-\frac {3}{2}\right )}{6} \]

[In]

int(x^2/(x^6 - 6*x^3 + 13),x)

[Out]

atan(x^3/2 - 3/2)/6