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

Optimal. Leaf size=72 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right ),\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}}+\frac{1}{3} \sqrt{x^4-1} x \]

[Out]

(x*Sqrt[-1 + x^4])/3 + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(3*Sq
rt[2]*Sqrt[-1 + x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0091386, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {321, 222} \[ \frac{1}{3} \sqrt{x^4-1} x+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[-1 + x^4],x]

[Out]

(x*Sqrt[-1 + x^4])/3 + (Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(3*Sq
rt[2]*Sqrt[-1 + x^4])

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2
)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 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{\sqrt{-1+x^2} \sqrt{1+x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{-1+x^2}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{-1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0092108, size = 44, normalized size = 0.61 \[ \frac{x \left (\sqrt{1-x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )+x^4-1\right )}{3 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[-1 + x^4],x]

[Out]

(x*(-1 + x^4 + Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4]))/(3*Sqrt[-1 + x^4])

________________________________________________________________________________________

Maple [C]  time = 0.007, size = 45, normalized size = 0.6 \begin{align*}{\frac{x}{3}\sqrt{{x}^{4}-1}}-{{\frac{i}{3}}{\it EllipticF} \left ( ix,i \right ) \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*I*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticF(I*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{x^{4}}{\sqrt{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(x^4/sqrt(x^4 - 1), x)

________________________________________________________________________________________

Sympy [C]  time = 0.788248, size = 27, normalized size = 0.38 \begin{align*} - \frac{i 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}} \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]

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