3.1.4 \(\int \frac {x}{\sqrt [3]{-1+x^2}} \, dx\) [4]

Optimal. Leaf size=13 \[ \frac {3}{4} \left (-1+x^2\right )^{2/3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \begin {gather*} \frac {3}{4} \left (x^2-1\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^2)^(1/3),x]

[Out]

(3*(-1 + x^2)^(2/3))/4

Rule 267

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {3}{4} \left (-1+x^2\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^2)^(1/3),x]

[Out]

(3*(-1 + x^2)^(2/3))/4

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Maple [A]
time = 0.25, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
default	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
trager	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
risch	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
gosper	\(\frac {3 \left (1+x \right ) \left (-1+x \right )}{4 \left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(16\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \left [2\right ], x^{2}\right )}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

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Fricas [A]
time = 0.34, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} \frac {3 \left (x^{2} - 1\right )^{\frac {2}{3}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(x**2 - 1)**(2/3)/4

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Giac [A]
time = 0.41, size = 9, normalized size = 0.69 \begin {gather*} \frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

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Mupad [B]
time = 0.11, size = 9, normalized size = 0.69 \begin {gather*} \frac {3\,{\left (x^2-1\right )}^{2/3}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 - 1)^(1/3),x)

[Out]

(3*(x^2 - 1)^(2/3))/4

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