3.10.6 \(\int \frac {x^4}{(1-x^4)^{3/2}} \, dx\) [906]

Optimal. Leaf size=25 \[ \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-1/2*EllipticF(x,I)+1/2*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 = {294, 227} \begin {gather*} \frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} F(\text {ArcSin}(x)|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(1 - x^4)^(3/2),x]

[Out]

x/(2*Sqrt[1 - x^4]) - EllipticF[ArcSin[x], -1]/2

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 294

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

Rubi steps

\begin {align*} \int \frac {x^4}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=\frac {x}{2 \sqrt {1-x^4}}-\frac {1}{2} 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 = 3.64, size = 32, normalized size = 1.28 \begin {gather*} \frac {1}{2} x \left (\frac {1}{\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/(1 - x^4)^(3/2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)

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Fricas [A]
time = 0.08, size = 31, normalized size = 1.24 \begin {gather*} -\frac {{\left (x^{4} - 1\right )} F(\arcsin \left (x\right )\,|\,-1) + \sqrt {-x^{4} + 1} x}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*((x^4 - 1)*elliptic_f(arcsin(x), -1) + sqrt(-x^4 + 1)*x)/(x^4 - 1)

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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 = 0.34, size = 31, normalized size = 1.24 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \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)**(3/2),x)

[Out]

x**5*gamma(5/4)*hyper((5/4, 3/2), (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)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^4}{{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(1 - x^4)^(3/2),x)

[Out]

int(x^4/(1 - x^4)^(3/2), x)

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