\(\int \frac {1}{x^2 (-1+x^4)^{3/4}} \, dx\) [26]

Optimal result

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

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{x^4-1}}{x} \]

[In]

Int[1/(x^2*(-1 + x^4)^(3/4)),x]

[Out]

(-1 + x^4)^(1/4)/x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-1+x^4}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{-1+x^4}}{x} \]

[In]

Integrate[1/(x^2*(-1 + x^4)^(3/4)),x]

[Out]

(-1 + x^4)^(1/4)/x

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
trager	\(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\)	\(12\)
risch	\(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\)	\(12\)
pseudoelliptic	\(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\)	\(12\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}{x \left (x^{4}-1\right )^{\frac {3}{4}}}\)	\(23\)
meijerg	\(-\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {3}{4}} \left (-x^{4}+1\right )^{\frac {1}{4}}}{\operatorname {signum}\left (x^{4}-1\right )^{\frac {3}{4}} x}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

(x^4 - 1)^(1/4)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.69 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\begin {cases} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\sqrt [4]{1 - \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-(-1 + x**(-4))**(1/4)*exp(I*pi/4)*gamma(-1/4)/(4*gamma(3/4)), 1/Abs(x**4) > 1), (-(1 - 1/x**4)**(1
/4)*gamma(-1/4)/(4*gamma(3/4)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

(x^4 - 1)^(1/4)/x

Giac [F]

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

[In]

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

[Out]

integrate(1/((x^4 - 1)^(3/4)*x^2), x)

Mupad [B] (verification not implemented)

Time = 5.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^4-1\right )}^{1/4}}{x} \]

[In]

int(1/(x^2*(x^4 - 1)^(3/4)),x)

[Out]

(x^4 - 1)^(1/4)/x