3.7.84 \(\int \frac {x}{x^2-\sqrt [3]{x^2}} \, dx\) [684]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {6847, 1607, 266} \begin {gather*} \frac {3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(x^2 - (x^2)^(1/3)),x]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

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]

Rule 6847

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.81 \begin {gather*} \frac {3}{4} \log \left (-1+\sqrt [3]{x^2}\right )+\frac {3}{4} \log \left (1+\sqrt [3]{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(x^2 - (x^2)^(1/3)),x]

[Out]

(3*Log[-1 + (x^2)^(1/3)])/4 + (3*Log[1 + (x^2)^(1/3)])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(12)=24\).
time = 0.41, size = 70, normalized size = 4.38

method	result	size
meijerg	\(\frac {3 \ln \left (1-\frac {x^{2}}{\left (x^{2}\right )^{\frac {1}{3}}}\right )}{4}\)	\(16\)
derivativedivides	\(\frac {\ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}-1\right )}{2}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}+\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}-\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}+\frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{2}\)	\(70\)
default	\(\frac {\ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}-1\right )}{2}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}+\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}-\frac {\ln \left (\left (x^{2}\right )^{\frac {2}{3}}-\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{4}+\frac {\ln \left (\left (x^{2}\right )^{\frac {1}{3}}+1\right )}{2}\)	\(70\)
trager	\(-\frac {\ln \left (-\frac {x^{8}+3 \left (x^{2}\right )^{\frac {1}{3}} x^{6}+6 \left (x^{2}\right )^{\frac {2}{3}} x^{4}+7 x^{4}+6 x^{2} \left (x^{2}\right )^{\frac {1}{3}}+3 \left (x^{2}\right )^{\frac {2}{3}}+1}{\left (x^{2}+1\right )^{3} \left (1+x \right )^{3} \left (-1+x \right )^{3}}\right )}{4}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.27, size = 21, normalized size = 1.31 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
time = 0.39, size = 32, normalized size = 2.00 \begin {gather*} -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 {gather*}

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 = 0.08, size = 19, normalized size = 1.19 \begin {gather*} - \frac {\log {\left (x \right )}}{2} + \frac {3 \log {\left (x^{2} - \sqrt [3]{x^{2}} \right )}}{4} \end {gather*}

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 = 1.88, size = 16, normalized size = 1.00 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 3.34, size = 10, normalized size = 0.62 \begin {gather*} \frac {3\,\ln \left ({\left (x^2\right )}^{2/3}-1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/((x^2)^(1/3) - x^2),x)

[Out]

(3*log((x^2)^(2/3) - 1))/4

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