3.9.88 \(\int \frac {1}{x^8 \sqrt {1-x^4}} \, dx\) [888]

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 = {331, 227} \begin {gather*} \frac {5}{21} F(\text {ArcSin}(x)|-1)-\frac {\sqrt {1-x^4}}{7 x^7}-\frac {5 \sqrt {1-x^4}}{21 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*Sqrt[1 - x^4]),x]

[Out]

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

Rule 227

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 331

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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 20, normalized size = 0.44 \begin {gather*} -\frac {\, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};x^4\right )}{7 x^7} \end {gather*}

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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Maple [A]
time = 0.16, size = 61, normalized size = 1.36

method	result	size
meijerg	\(-\frac {\hypergeom \left (\left [-\frac {7}{4}, \frac {1}{2}\right ], \left [-\frac {3}{4}\right ], x^{4}\right )}{7 x^{7}}\)	\(15\)
risch	\(\frac {5 x^{8}-2 x^{4}-3}{21 x^{7} \sqrt {-x^{4}+1}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(59\)
default	\(-\frac {\sqrt {-x^{4}+1}}{7 x^{7}}-\frac {5 \sqrt {-x^{4}+1}}{21 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(61\)
elliptic	\(-\frac {\sqrt {-x^{4}+1}}{7 x^{7}}-\frac {5 \sqrt {-x^{4}+1}}{21 x^{3}}+\frac {5 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{21 \sqrt {-x^{4}+1}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.08, size = 33, normalized size = 0.73 \begin {gather*} \frac {5 \, x^{7} F(\arcsin \left (x\right )\,|\,-1) - {\left (5 \, x^{4} + 3\right )} \sqrt {-x^{4} + 1}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(5*x^7*elliptic_f(arcsin(x), -1) - (5*x^4 + 3)*sqrt(-x^4 + 1))/x^7

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Sympy [A]
time = 0.48, size = 37, normalized size = 0.82 \begin {gather*} \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 )} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^8\,\sqrt {1-x^4}} \,d x \end {gather*}

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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