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

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (1-x^{2/3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 266} \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (1-x^{2/3}\right ) \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \log \left (-1+\sqrt [3]{x}\right )+\frac {3}{2} \log \left (1+\sqrt [3]{x}\right ) \]

[In]

Integrate[(-x^(1/3) + x)^(-1),x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {2}{3}} - 1\right ) \]

[In]

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

[Out]

3/2*log(x^(2/3) - 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3 \log {\left (\sqrt [3]{x} - 1 \right )}}{2} + \frac {3 \log {\left (\sqrt [3]{x} + 1 \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \log \left (x^{\frac {1}{3}} - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3}{2} \, \log \left (x^{\frac {1}{3}} + 1\right ) + \frac {3}{2} \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{-\sqrt [3]{x}+x} \, dx=\frac {3\,\ln \left (x^{2/3}-1\right )}{2} \]

[In]

int(1/(x - x^(1/3)),x)

[Out]

(3*log(x^(2/3) - 1))/2