3.3.14 \(\int \frac {1}{x^6 (-1+x^4)^{3/4}} \, dx\)

Optimal. Leaf size=23 \[ \frac {\sqrt [4]{x^4-1} \left (4 x^4+1\right )}{5 x^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \begin {gather*} \frac {4 \sqrt [4]{x^4-1}}{5 x}+\frac {\sqrt [4]{x^4-1}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(-1 + x^4)^(3/4)),x]

[Out]

(-1 + x^4)^(1/4)/(5*x^5) + (4*(-1 + x^4)^(1/4))/(5*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]

Rule 271

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

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx &=\frac {\sqrt [4]{-1+x^4}}{5 x^5}+\frac {4}{5} \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{-1+x^4}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^4}}{5 x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{x^4-1} \left (4 x^4+1\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(-1 + x^4)^(3/4)),x]

[Out]

((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5)

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IntegrateAlgebraic [A]  time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^6*(-1 + x^4)^(3/4)),x]

[Out]

((-1 + x^4)^(1/4)*(1 + 4*x^4))/(5*x^5)

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fricas [A]  time = 0.45, size = 19, normalized size = 0.83 \begin {gather*} \frac {{\left (4 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4-1)^(3/4),x, algorithm="fricas")

[Out]

1/5*(4*x^4 + 1)*(x^4 - 1)^(1/4)/x^5

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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^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4-1)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((x^4 - 1)^(3/4)*x^6), x)

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maple [A]  time = 0.09, size = 20, normalized size = 0.87

method	result	size
trager	\(\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (4 x^{4}+1\right )}{5 x^{5}}\)	\(20\)
risch	\(\frac {4 x^{8}-3 x^{4}-1}{5 \left (x^{4}-1\right )^{\frac {3}{4}} x^{5}}\)	\(25\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (4 x^{4}+1\right )}{5 x^{5} \left (x^{4}-1\right )^{\frac {3}{4}}}\)	\(31\)
meijerg	\(-\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} \left (4 x^{4}+1\right ) \left (-x^{4}+1\right )^{\frac {1}{4}}}{5 \mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}} x^{5}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^4-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/5*(x^4-1)^(1/4)*(4*x^4+1)/x^5

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maxima [A]  time = 0.32, size = 24, normalized size = 1.04 \begin {gather*} \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - \frac {{\left (x^{4} - 1\right )}^{\frac {5}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4-1)^(3/4),x, algorithm="maxima")

[Out]

(x^4 - 1)^(1/4)/x - 1/5*(x^4 - 1)^(5/4)/x^5

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mupad [B]  time = 0.22, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (x^4-1\right )}^{1/4}+4\,x^4\,{\left (x^4-1\right )}^{1/4}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^4 - 1)^(3/4)),x)

[Out]

((x^4 - 1)^(1/4) + 4*x^4*(x^4 - 1)^(1/4))/(5*x^5)

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sympy [B]  time = 0.82, size = 124, normalized size = 5.39 \begin {gather*} \begin {cases} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{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 {5}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{1 - \frac {1}{x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{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**6/(x**4-1)**(3/4),x)

[Out]

Piecewise((-(-1 + x**(-4))**(1/4)*exp(-3*I*pi/4)*gamma(-5/4)/(4*gamma(3/4)) - (-1 + x**(-4))**(1/4)*exp(-3*I*p
i/4)*gamma(-5/4)/(16*x**4*gamma(3/4)), 1/Abs(x**4) > 1), ((1 - 1/x**4)**(1/4)*gamma(-5/4)/(4*gamma(3/4)) + (1
- 1/x**4)**(1/4)*gamma(-5/4)/(16*x**4*gamma(3/4)), True))

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