3.2959 \(\int \frac{1}{1+(x^3)^{2/3}} \, dx\)

Optimal. Leaf size=17 \[ \frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \]

[Out]

(x*ArcTan[(x^3)^(1/3)])/(x^3)^(1/3)

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Rubi [A]  time = 0.0049069, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {254, 203} \[ \frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + (x^3)^(2/3))^(-1),x]

[Out]

(x*ArcTan[(x^3)^(1/3)])/(x^3)^(1/3)

Rule 254

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)]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{1+\left (x^3\right )^{2/3}} \, dx &=\frac{x \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}}\\ &=\frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0085616, size = 17, normalized size = 1. \[ \frac{x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + (x^3)^(2/3))^(-1),x]

[Out]

(x*ArcTan[(x^3)^(1/3)])/(x^3)^(1/3)

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Maple [A]  time = 0.051, size = 14, normalized size = 0.8 \begin{align*}{x\arctan \left ( \sqrt [3]{{x}^{3}} \right ){\frac{1}{\sqrt [3]{{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+(x^3)^(2/3)),x)

[Out]

x*arctan((x^3)^(1/3))/(x^3)^(1/3)

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Maxima [A]  time = 1.43102, size = 3, normalized size = 0.18 \begin{align*} \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(x^3)^(2/3)),x, algorithm="maxima")

[Out]

arctan(x)

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Fricas [A]  time = 1.34998, size = 28, normalized size = 1.65 \begin{align*} \arctan \left ({\left (x^{3}\right )}^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(x^3)^(2/3)),x, algorithm="fricas")

[Out]

arctan((x^3)^(1/3))

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Sympy [A]  time = 0.091499, size = 2, normalized size = 0.12 \begin{align*} \operatorname{atan}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(x**3)**(2/3)),x)

[Out]

atan(x)

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Giac [A]  time = 1.18994, size = 3, normalized size = 0.18 \begin{align*} \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(x^3)^(2/3)),x, algorithm="giac")

[Out]

arctan(x)