3.888 \(\int \frac {1}{x^8 \sqrt {1-x^4}} \, dx\)

Optimal. Leaf size=45 \[ -\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

5/21*EllipticF(x,I)-1/7*(-x^4+1)^(1/2)/x^7-5/21*(-x^4+1)^(1/2)/x^3

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {325, 221} \[ -\frac {5 \sqrt {1-x^4}}{21 x^3}-\frac {\sqrt {1-x^4}}{7 x^7}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(7*x^7) - (5*Sqrt[1 - x^4])/(21*x^3) + (5*EllipticF[ArcSin[x], -1])/21

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{7 x^7}+\frac {5}{7} \int \frac {1}{x^4 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3}+\frac {5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}

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Mathematica [C]  time = 0.00, size = 20, normalized size = 0.44 \[ -\frac {\, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};x^4\right )}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*Sqrt[1 - x^4]),x]

[Out]

-1/7*Hypergeometric2F1[-7/4, 1/2, -3/4, x^4]/x^7

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fricas [F]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 1}}{x^{12} - x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 1)/(x^12 - x^8), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^8), x)

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maple [A]  time = 0.01, size = 61, normalized size = 1.36 \[ \frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}-\frac {5 \sqrt {-x^{4}+1}}{21 x^{3}}-\frac {\sqrt {-x^{4}+1}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(-x^4+1)^(1/2),x)

[Out]

-1/7*(-x^4+1)^(1/2)/x^7-5/21*(-x^4+1)^(1/2)/x^3+5/21*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*EllipticF(x,I
)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^8), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^8\,\sqrt {1-x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(1 - x^4)^(1/2)),x)

[Out]

int(1/(x^8*(1 - x^4)^(1/2)), x)

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sympy [A]  time = 1.96, size = 37, normalized size = 0.82 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(-x**4+1)**(1/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), x**4*exp_polar(2*I*pi))/(4*x**7*gamma(-3/4))

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