3.9.85 \(\int \frac {x^4}{\sqrt {1-x^4}} \, dx\) [885]

Optimal. Leaf size=25 \[ -\frac {1}{3} x \sqrt {1-x^4}+\frac {1}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, 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 = {327, 227} \begin {gather*} \frac {1}{3} F(\text {ArcSin}(x)|-1)-\frac {1}{3} x \sqrt {1-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[1 - x^4],x]

[Out]

-1/3*(x*Sqrt[1 - x^4]) + 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 327

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

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {1-x^4}} \, dx &=-\frac {1}{3} x \sqrt {1-x^4}+\frac {1}{3} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-\frac {1}{3} x \sqrt {1-x^4}+\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 = 10.02, size = 32, normalized size = 1.28 \begin {gather*} \frac {1}{3} x \left (-\sqrt {1-x^4}+\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[1 - x^4],x]

[Out]

(x*(-Sqrt[1 - x^4] + Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/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. 44 vs. \(2 (19 ) = 38\).
time = 0.16, size = 45, normalized size = 1.80

method	result	size
meijerg	\(\frac {x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], x^{4}\right )}{5}\)	\(15\)
default	\(-\frac {x \sqrt {-x^{4}+1}}{3}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(45\)
elliptic	\(-\frac {x \sqrt {-x^{4}+1}}{3}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(45\)
risch	\(\frac {x \left (x^{4}-1\right )}{3 \sqrt {-x^{4}+1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{3 \sqrt {-x^{4}+1}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^4/sqrt(-x^4 + 1), x)

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Fricas [A]
time = 0.07, size = 12, normalized size = 0.48 \begin {gather*} -\frac {1}{3} \, \sqrt {-x^{4} + 1} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(-x^4 + 1)*x

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
time = 0.33, size = 31, normalized size = 1.24 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), x**4*exp_polar(2*I*pi))/(4*gamma(9/4))

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

[Out]

integrate(x^4/sqrt(-x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(1 - x^4)^(1/2),x)

[Out]

int(x^4/(1 - x^4)^(1/2), x)

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