3.2386 \(\int \frac{1}{(1+x^{2/3}) x^{2/3}} \, dx\)

Optimal. Leaf size=8 \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0038923, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {341, 203} \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^(2/3))*x^(2/3)),x]

[Out]

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

Rule 341

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

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

Mathematica [A]  time = 0.0019225, size = 8, normalized size = 1. \[ 3 \tan ^{-1}\left (\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^(2/3))*x^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \begin{align*} 3\,\arctan \left ( \sqrt [3]{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46017, size = 8, normalized size = 1. \begin{align*} 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^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.4807, size = 26, normalized size = 3.25 \begin{align*} 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^(2/3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.261398, size = 7, normalized size = 0.88 \begin{align*} 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**(2/3),x)

[Out]

3*atan(x**(1/3))

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Giac [A]  time = 1.17291, size = 8, normalized size = 1. \begin{align*} 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^(2/3),x, algorithm="giac")

[Out]

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