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

Optimal. Leaf size=74 \[ \frac{\sqrt{x^2-1} \sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right ),\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}}+\frac{\sqrt{x^4-1}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0093243, antiderivative size = 74, 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 = {325, 222} \[ \frac{\sqrt{x^4-1}}{3 x^3}+\frac{\sqrt{x^2-1} \sqrt{x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[-1 + x^4]),x]

[Out]

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

Rule 325

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]

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{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{\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 = 0.0050697, size = 40, normalized size = 0.54 \[ -\frac{\sqrt{1-x^4} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};x^4\right )}{3 x^3 \sqrt{x^4-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-1 + x^4]),x]

[Out]

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

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Maple [C]  time = 0.009, size = 47, normalized size = 0.6 \begin{align*}{\frac{1}{3\,{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(1/x^4/(x^4-1)^(1/2),x)

[Out]

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

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} - 1}}{x^{8} - x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^4 - 1)/(x^8 - x^4), x)

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Sympy [C]  time = 0.965629, size = 31, normalized size = 0.42 \begin{align*} - \frac{i \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}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4)/(4*x**3*gamma(1/4))

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

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)