\(\int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx\) [3036]

Optimal result

Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {x \arctan \left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {260, 209} \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {x \arctan \left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

[In]

Int[(1 + 4*(x^6)^(1/3))^(-1),x]

[Out]

(x*ArcTan[2*(x^6)^(1/6)])/(2*(x^6)^(1/6))

Rule 209

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

Rule 260

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\sqrt [6]{x^6}\right )}{\sqrt [6]{x^6}} \\ & = \frac {x \tan ^{-1}\left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {x \arctan \left (2 \sqrt [6]{x^6}\right )}{2 \sqrt [6]{x^6}} \]

[In]

Integrate[(1 + 4*(x^6)^(1/3))^(-1),x]

[Out]

(x*ArcTan[2*(x^6)^(1/6)])/(2*(x^6)^(1/6))

Maple [A] (verified)

Time = 5.98 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
meijerg	\(\frac {x \arctan \left (2 \left (x^{6}\right )^{\frac {1}{6}}\right )}{2 \left (x^{6}\right )^{\frac {1}{6}}}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {1}{2} \, \arctan \left (2 \, x\right ) \]

[In]

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

[Out]

1/2*arctan(2*x)

Sympy [F]

\[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\int \frac {1}{4 \sqrt [3]{x^{6}} + 1}\, dx \]

[In]

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

[Out]

Integral(1/(4*(x**6)**(1/3) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {1}{2} \, \arctan \left (2 \, x\right ) \]

[In]

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

[Out]

1/2*arctan(2*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\frac {1}{2} \, \arctan \left (2 \, x\right ) \]

[In]

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

[Out]

1/2*arctan(2*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{1+4 \sqrt [3]{x^6}} \, dx=\int \frac {1}{4\,{\left (x^6\right )}^{1/3}+1} \,d x \]

[In]

int(1/(4*(x^6)^(1/3) + 1),x)

[Out]

int(1/(4*(x^6)^(1/3) + 1), x)