3.6.70 \(\int \frac {1}{-\sqrt [3]{x}+x^{2/3}} \, dx\) [570]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 272, 45} \begin {gather*} 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \end {gather*}

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 45

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])

Rule 272

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 1607

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]

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 \text {Subst}\left (\int \frac {x}{-1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} 3 \sqrt [3]{x}+3 \log \left (-1+\sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x^(1/3) + 3*Log[-1 + x^(1/3)]

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Maple [A]
time = 0.20, size = 15, normalized size = 0.75

method	result	size
derivativedivides	\(3 x^{\frac {1}{3}}+3 \ln \left (-1+x^{\frac {1}{3}}\right )\)	\(15\)
default	\(3 x^{\frac {1}{3}}+3 \ln \left (-1+x^{\frac {1}{3}}\right )\)	\(15\)
meijerg	\(3 x^{\frac {1}{3}}+3 \ln \left (1-x^{\frac {1}{3}}\right )\)	\(17\)
trager	\(3 x^{\frac {1}{3}}+\ln \left (-3 x^{\frac {2}{3}}+3 x^{\frac {1}{3}}+x -1\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.70 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} - 1\right ) \end {gather*}

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 = 0.34, size = 14, normalized size = 0.70 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} - 1\right ) \end {gather*}

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 = 0.05, size = 15, normalized size = 0.75 \begin {gather*} 3 \sqrt [3]{x} + 3 \log {\left (\sqrt [3]{x} - 1 \right )} \end {gather*}

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 = 4.53, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.08, size = 14, normalized size = 0.70 \begin {gather*} 3\,\ln \left (x^{1/3}-1\right )+3\,x^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^(1/3) - x^(2/3)),x)

[Out]

3*log(x^(1/3) - 1) + 3*x^(1/3)

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