3.2384 \(\int \frac{1}{1+x^{2/3}} \, dx\)

Optimal. Leaf size=16 \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

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Rubi [A]  time = 0.0053561, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {243, 321, 203} \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(2/3))^(-1),x]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

Rule 243

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(a + b*
x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, p}, x] && FractionQ[n]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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

Mathematica [A]  time = 0.0036034, size = 16, normalized size = 1. \[ 3 \sqrt [3]{x}-3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(2/3))^(-1),x]

[Out]

3*x^(1/3) - 3*ArcTan[x^(1/3)]

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Maple [B]  time = 0.028, size = 41, normalized size = 2.6 \begin{align*} \arctan \left ( x \right ) +3\,\sqrt [3]{x}-2\,\arctan \left ( \sqrt [3]{x} \right ) -\arctan \left ( 2\,\sqrt [3]{x}-\sqrt{3} \right ) -\arctan \left ( 2\,\sqrt [3]{x}+\sqrt{3} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44279, size = 16, normalized size = 1. \begin{align*} 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))

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Fricas [A]  time = 1.49598, size = 42, normalized size = 2.62 \begin{align*} 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))

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Sympy [A]  time = 0.150323, size = 14, normalized size = 0.88 \begin{align*} 3 \sqrt [3]{x} - 3 \operatorname{atan}{\left (\sqrt [3]{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(1/3) - 3*atan(x**(1/3))

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Giac [A]  time = 1.17359, size = 16, normalized size = 1. \begin{align*} 3 \, x^{\frac{1}{3}} - 3 \, \arctan \left (x^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) - 3*arctan(x^(1/3))