3.684 \(\int \frac{x}{x^2-\sqrt [3]{x^2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

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Rubi [A]  time = 0.061193, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6715, 1593, 260} \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(x^2 - (x^2)^(1/3)),x]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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{x}{x^2-\sqrt [3]{x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{x}+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (-1+x^{2/3}\right ) \sqrt [3]{x}} \, dx,x,x^2\right )\\ &=\frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0267944, size = 16, normalized size = 1. \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(x^2 - (x^2)^(1/3)),x]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

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Maple [B]  time = 0.033, size = 70, normalized size = 4.4 \begin{align*}{\frac{\ln \left ({x}^{2}-1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{1}{2}\ln \left ( \sqrt [3]{{x}^{2}}-1 \right ) }-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{x}^{2}}+1 \right ) }-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{x}^{2}}+1 \right ) }+{\frac{1}{2}\ln \left ( \sqrt [3]{{x}^{2}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(x^2-1)+1/4*ln(x^2+1)+1/2*ln((x^2)^(1/3)-1)-1/4*ln((x^2)^(2/3)+(x^2)^(1/3)+1)-1/4*ln((x^2)^(2/3)-(x^2)^(
1/3)+1)+1/2*ln((x^2)^(1/3)+1)

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Maxima [A]  time = 1.12492, size = 28, normalized size = 1.75 \begin{align*} \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} + 1\right ) + \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*log((x^2)^(1/3) + 1) + 3/4*log((x^2)^(1/3) - 1)

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Fricas [B]  time = 1.64885, size = 80, normalized size = 5. \begin{align*} -3 \, \log \left (\frac{{\left (x^{2}\right )}^{\frac{1}{3}}}{x}\right ) + \frac{3}{4} \, \log \left (-\frac{x^{2} -{\left (x^{2}\right )}^{\frac{1}{3}}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*log((x^2)^(1/3)/x) + 3/4*log(-(x^2 - (x^2)^(1/3))/x^2)

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Sympy [A]  time = 0.204456, size = 19, normalized size = 1.19 \begin{align*} - \frac{\log{\left (x \right )}}{2} + \frac{3 \log{\left (x^{2} - \sqrt [3]{x^{2}} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x)/2 + 3*log(x**2 - (x**2)**(1/3))/4

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Giac [A]  time = 1.14409, size = 22, normalized size = 1.38 \begin{align*} \frac{3}{4} \, \log \left ({\left | \left (x \mathrm{sgn}\left (x\right )\right )^{\frac{1}{3}} x \mathrm{sgn}\left (x\right ) - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*log(abs((x*sgn(x))^(1/3)*x*sgn(x) - 1))