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

Optimal. Leaf size=13 \[ \frac {\sqrt [4]{x^4-1}}{x} \]

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} \frac {\sqrt [4]{x^4-1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(-1 + x^4)^(3/4)),x]

[Out]

(-1 + x^4)^(1/4)/x

Rule 264

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

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{x^4-1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(-1 + x^4)^(3/4)),x]

[Out]

(-1 + x^4)^(1/4)/x

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IntegrateAlgebraic [A]  time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^4)^(1/4)/x

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fricas [A]  time = 0.46, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4 - 1)^(1/4)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 - 1)^(3/4)*x^2), x)

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maple [A]  time = 0.08, size = 12, normalized size = 0.92

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4 - 1)^(1/4)/x

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mupad [B]  time = 0.21, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\left (x^4-1\right )}^{1/4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^4 - 1)^(3/4)),x)

[Out]

(x^4 - 1)^(1/4)/x

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sympy [B]  time = 0.58, size = 61, normalized size = 4.69 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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