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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, 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} \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{x^2-1} \]

[In]

Int[x/(-1 + x^2)^(3/4),x]

[Out]

2*(-1 + x^2)^(1/4)

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}& = 2 \sqrt [4]{-1+x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{-1+x^2} \]

[In]

Integrate[x/(-1 + x^2)^(3/4),x]

[Out]

2*(-1 + x^2)^(1/4)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91

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

[In]

int(x/(x^2-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="fricas")

[Out]

2*(x^2 - 1)^(1/4)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{x^{2} - 1} \]

[In]

integrate(x/(x**2-1)**(3/4),x)

[Out]

2*(x**2 - 1)**(1/4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="maxima")

[Out]

2*(x^2 - 1)^(1/4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="giac")

[Out]

2*(x^2 - 1)^(1/4)

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2\,{\left (x^2-1\right )}^{1/4} \]

[In]

int(x/(x^2 - 1)^(3/4),x)

[Out]

2*(x^2 - 1)^(1/4)