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

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \end {gather*}

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 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])

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)]

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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {x \tan ^{-1}\left (\sqrt [3]{x^3}\right )}{\sqrt [3]{x^3}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 4.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{1+\left (x^3\right )^{2/3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + (x^3)^(2/3))^(-1),x]

[Out]

Defer[IntegrateAlgebraic][(1 + (x^3)^(2/3))^(-1), x]

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fricas [A]  time = 0.83, size = 6, normalized size = 0.35 \begin {gather*} \arctan \left ({\left (x^{3}\right )}^{\frac {1}{3}}\right ) \end {gather*}

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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giac [A]  time = 0.16, size = 2, normalized size = 0.12 \begin {gather*} \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x)

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maple [A]  time = 0.22, size = 14, normalized size = 0.82 \begin {gather*} \frac {x \arctan \left (\left (x^{3}\right )^{\frac {1}{3}}\right )}{\left (x^{3}\right )^{\frac {1}{3}}} \end {gather*}

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 = 1.17, size = 2, normalized size = 0.12 \begin {gather*} \arctan \relax (x) \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{{\left (x^3\right )}^{2/3}+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^3)^(2/3) + 1),x)

[Out]

int(1/((x^3)^(2/3) + 1), x)

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sympy [A]  time = 0.13, size = 2, normalized size = 0.12 \begin {gather*} \operatorname {atan}{\relax (x )} \end {gather*}

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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