3.885 \(\int \frac{x^4}{\sqrt{1-x^4}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0048619, antiderivative size = 25, normalized size of antiderivative = 1., 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 = {321, 221} \[ \frac{1}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{1}{3} x \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[1 - x^4],x]

[Out]

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

Rule 321

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]

Rule 221

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]

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*}

Mathematica [C]  time = 0.0060935, size = 32, normalized size = 1.28 \[ \frac{1}{3} x \left (\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )-\sqrt{1-x^4}\right ) \]

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

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 45, normalized size = 1.8 \begin{align*} -{\frac{x}{3}\sqrt{-{x}^{4}+1}}+{\frac{{\it EllipticF} \left ( x,i \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 1} x^{4}}{x^{4} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-x^4 + 1)*x^4/(x^4 - 1), x)

________________________________________________________________________________________

Sympy [B]  time = 0.815275, size = 31, normalized size = 1.24 \begin{align*} \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{align*}

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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{-x^{4} + 1}}\,{d x} \end{align*}

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)