3.3.26 \(\int \frac {1}{-\sqrt [3]{x}+x} \, dx\) [226]

Optimal. Leaf size=14 \[ \frac {3}{2} \log \left (1-x^{2/3}\right ) \]

[Out]

3/2*ln(1-x^(2/3))

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Rubi [A]
time = 0.00, antiderivative size = 14, 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 = {1607, 266} \begin {gather*} \frac {3}{2} \log \left (1-x^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^(1/3) + x)^(-1),x]

[Out]

(3*Log[1 - x^(2/3)])/2

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.79 \begin {gather*} \frac {3}{2} \log \left (-1+\sqrt [3]{x}\right )+\frac {3}{2} \log \left (1+\sqrt [3]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^(1/3) + x)^(-1),x]

[Out]

(3*Log[-1 + x^(1/3)])/2 + (3*Log[1 + x^(1/3)])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(10)=20\).
time = 0.10, size = 50, normalized size = 3.57

method	result	size
meijerg	\(\frac {3 \ln \left (1-x^{\frac {2}{3}}\right )}{2}\)	\(11\)
derivativedivides	\(\frac {3 \ln \left (-1+x^{\frac {1}{3}}\right )}{2}+\frac {3 \ln \left (x^{\frac {1}{3}}+1\right )}{2}\)	\(18\)
trager	\(\frac {\ln \left (3 x^{\frac {2}{3}}-3 x^{\frac {4}{3}}+x^{2}-1\right )}{2}\)	\(19\)
default	\(\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}+\ln \left (-1+x^{\frac {1}{3}}\right )-\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{2}+\ln \left (x^{\frac {1}{3}}+1\right )-\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.88, size = 17, normalized size = 1.21 \begin {gather*} \frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.50, size = 8, normalized size = 0.57 \begin {gather*} \frac {3}{2} \, \log \left (x^{\frac {2}{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*log(x^(2/3) - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
time = 0.06, size = 22, normalized size = 1.57 \begin {gather*} \frac {3 \log {\left (\sqrt [3]{x} - 1 \right )}}{2} + \frac {3 \log {\left (\sqrt [3]{x} + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x**(1/3) - 1)/2 + 3*log(x**(1/3) + 1)/2

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Giac [A]
time = 0.45, size = 18, normalized size = 1.29 \begin {gather*} \frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \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),x, algorithm="giac")

[Out]

3/2*log(x^(1/3) + 1) + 3/2*log(abs(x^(1/3) - 1))

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Mupad [B]
time = 0.15, size = 8, normalized size = 0.57 \begin {gather*} \frac {3\,\ln \left (x^{2/3}-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^(1/3)),x)

[Out]

(3*log(x^(2/3) - 1))/2

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