3.9.93 \(\int \frac {1}{x^6 \sqrt {1-x^4}} \, dx\) [893]

Optimal. Leaf size=53 \[ -\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {331, 313, 227, 1195, 435} \begin {gather*} \frac {3}{5} F(\text {ArcSin}(x)|-1)-\frac {3}{5} E(\text {ArcSin}(x)|-1)-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {\sqrt {1-x^4}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*Sqrt[1 - x^4]),x]

[Out]

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

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 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

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]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{5 x^5}+\frac {3}{5} \int \frac {1}{x^2 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} \int \frac {x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}+\frac {3}{5} \int \frac {1}{\sqrt {1-x^4}} \, dx-\frac {3}{5} \int \frac {1+x^2}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}+\frac {3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {3}{5} \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{5 x^5}-\frac {3 \sqrt {1-x^4}}{5 x}-\frac {3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {3}{5} 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 = 10.02, size = 20, normalized size = 0.38 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};x^4\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*Sqrt[1 - x^4]),x]

[Out]

-1/5*Hypergeometric2F1[-5/4, 1/2, -1/4, x^4]/x^5

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Maple [A]
time = 0.16, size = 68, normalized size = 1.28

method	result	size
meijerg	\(-\frac {\hypergeom \left (\left [-\frac {5}{4}, \frac {1}{2}\right ], \left [-\frac {1}{4}\right ], x^{4}\right )}{5 x^{5}}\)	\(15\)
risch	\(\frac {3 x^{8}-2 x^{4}-1}{5 x^{5} \sqrt {-x^{4}+1}}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(66\)
default	\(-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {3 \sqrt {-x^{4}+1}}{5 x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(68\)
elliptic	\(-\frac {\sqrt {-x^{4}+1}}{5 x^{5}}-\frac {3 \sqrt {-x^{4}+1}}{5 x}+\frac {3 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{5 \sqrt {-x^{4}+1}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(-x^4+1)^(1/2)/x^5-3/5*(-x^4+1)^(1/2)/x+3/5*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I)-E
llipticE(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^6/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^6), x)

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Fricas [A]
time = 0.08, size = 41, normalized size = 0.77 \begin {gather*} -\frac {3 \, x^{5} E(\arcsin \left (x\right )\,|\,-1) - 3 \, x^{5} F(\arcsin \left (x\right )\,|\,-1) + {\left (3 \, x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.43, size = 37, normalized size = 0.70 \begin {gather*} \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(-x**4+1)**(1/2),x)

[Out]

gamma(-5/4)*hyper((-5/4, 1/2), (-1/4,), x**4*exp_polar(2*I*pi))/(4*x**5*gamma(-1/4))

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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^6/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^6\,\sqrt {1-x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(1 - x^4)^(1/2)),x)

[Out]

int(1/(x^6*(1 - x^4)^(1/2)), x)

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