3.9.87 \(\int \frac {1}{x^4 \sqrt {1-x^4}} \, dx\) [887]

Optimal. Leaf size=27 \[ -\frac {\sqrt {1-x^4}}{3 x^3}+\frac {1}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, 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 {1}{3} F(\text {ArcSin}(x)|-1)-\frac {\sqrt {1-x^4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[1 - x^4]),x]

[Out]

-1/3*Sqrt[1 - x^4]/x^3 + EllipticF[ArcSin[x], -1]/3

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^4 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{3 x^3}+\frac {1}{3} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{3 x^3}+\frac {1}{3} 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 = 0.74 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};x^4\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[1 - x^4]),x]

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21 ) = 42\).
time = 0.16, size = 47, normalized size = 1.74

method	result	size
meijerg	\(-\frac {\hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], x^{4}\right )}{3 x^{3}}\)	\(15\)
default	\(-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(47\)
elliptic	\(-\frac {\sqrt {-x^{4}+1}}{3 x^{3}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(47\)
risch	\(\frac {x^{4}-1}{3 x^{3} \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^4), x)

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Fricas [A]
time = 0.08, size = 25, normalized size = 0.93 \begin {gather*} \frac {x^{3} F(\arcsin \left (x\right )\,|\,-1) - \sqrt {-x^{4} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.37, size = 34, normalized size = 1.26 \begin {gather*} \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4*exp_polar(2*I*pi))/(4*x**3*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^4/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^4\,\sqrt {1-x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(1 - x^4)^(1/2)),x)

[Out]

int(1/(x^4*(1 - x^4)^(1/2)), x)

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