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

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \left (-1+x^2\right )^{2/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \left (x^2-1\right )^{2/3} \]

[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*} \text {integral}& = \frac {3}{4} \left (-1+x^2\right )^{2/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \left (-1+x^2\right )^{2/3} \]

[In]

Integrate[x/(-1 + x^2)^(1/3),x]

[Out]

(3*(-1 + x^2)^(2/3))/4

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 10, normalized size of antiderivative = 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	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
pseudoelliptic	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
gosper	\(\frac {3 \left (x -1\right ) \left (1+x \right )}{4 \left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(16\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1\right ], \left [2\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \]

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3 \left (x^{2} - 1\right )^{\frac {2}{3}}}{4} \]

[In]

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

[Out]

3*(x**2 - 1)**(2/3)/4

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \]

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3}{4} \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} \]

[In]

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

[Out]

3/4*(x^2 - 1)^(2/3)

Mupad [B] (verification not implemented)

Time = 5.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt [3]{-1+x^2}} \, dx=\frac {3\,{\left (x^2-1\right )}^{2/3}}{4} \]

[In]

int(x/(x^2 - 1)^(1/3),x)

[Out]

(3*(x^2 - 1)^(2/3))/4