### 3.226 $$\int \frac{1}{-\sqrt {x}+x} \, dx$$

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

[Out]

(3*Log[1 - x^(2/3)])/2

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Rubi [A]  time = 0.00433, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {1593, 260} $\frac{3}{2} \log \left (1-x^{2/3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(-x^(1/3) + x)^(-1),x]

[Out]

(3*Log[1 - x^(2/3)])/2

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 260

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]

Rubi steps

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

Mathematica [A]  time = 0.0018832, size = 14, normalized size = 1. $\frac{3}{2} \log \left (1-x^{2/3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-x^(1/3) + x)^(-1),x]

[Out]

(3*Log[1 - x^(2/3)])/2

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Maple [B]  time = 0.011, size = 50, normalized size = 3.6 \begin{align*}{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}+\ln \left ( -1+\sqrt {x} \right ) -{\frac{1}{2}\ln \left ({x}^{{\frac{2}{3}}}+\sqrt {x}+1 \right ) }+\ln \left ( \sqrt {x}+1 \right ) -{\frac{1}{2}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt {x}+1 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.927172, size = 23, normalized size = 1.64 \begin{align*} \frac{3}{2} \, \log \left (x^{\frac{1}{3}} + 1\right ) + \frac{3}{2} \, \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),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 = 1.90974, size = 30, normalized size = 2.14 \begin{align*} \frac{3}{2} \, \log \left (x^{\frac{2}{3}} - 1\right ) \end{align*}

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]  time = 0.178474, size = 22, normalized size = 1.57 \begin{align*} \frac{3 \log{\left (\sqrt {x} - 1 \right )}}{2} + \frac{3 \log{\left (\sqrt {x} + 1 \right )}}{2} \end{align*}

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 = 1.05442, size = 24, normalized size = 1.71 \begin{align*} \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{align*}

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))