3.570 \(\int \frac{1}{-\sqrt [3]{x}+x^{2/3}} \, dx\)

Optimal. Leaf size=20 \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]

[Out]

3*x^(1/3) + 3*Log[1 - x^(1/3)]

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Rubi [A]  time = 0.0091215, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1593, 266, 43} \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x^(1/3) + x^(2/3))^(-1),x]

[Out]

3*x^(1/3) + 3*Log[1 - x^(1/3)]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.006239, size = 18, normalized size = 0.9 \[ 3 \left (\sqrt [3]{x}+\log \left (1-\sqrt [3]{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*(x^(1/3) + Log[1 - x^(1/3)])

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*} 3\,\sqrt [3]{x}+3\,\ln \left ( \sqrt [3]{x}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3)+3*ln(x^(1/3)-1)

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Maxima [A]  time = 1.27376, size = 19, normalized size = 0.95 \begin{align*} 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) + 3*log(x^(1/3) - 1)

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Fricas [A]  time = 1.28871, size = 43, normalized size = 2.15 \begin{align*} 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) + 3*log(x^(1/3) - 1)

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Sympy [A]  time = 0.137258, size = 15, normalized size = 0.75 \begin{align*} 3 \sqrt [3]{x} + 3 \log{\left (\sqrt [3]{x} - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(1/3) + 3*log(x**(1/3) - 1)

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Giac [A]  time = 1.10568, size = 20, normalized size = 1. \begin{align*} 3 \, x^{\frac{1}{3}} + 3 \, \log \left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x^(1/3) + 3*log(abs(x^(1/3) - 1))