3.973 \(\int \frac{1}{x^4 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{96} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right )-\frac{\sqrt{16-x^4}}{48 x^3} \]

[Out]

-Sqrt[16 - x^4]/(48*x^3) + EllipticF[ArcSin[x/2], -1]/96

________________________________________________________________________________________

Rubi [A]  time = 0.0055191, antiderivative size = 31, 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 = {325, 221} \[ \frac{1}{96} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{\sqrt{16-x^4}}{48 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(48*x^3) + EllipticF[ArcSin[x/2], -1]/96

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 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{1}{x^4 \sqrt{16-x^4}} \, dx &=-\frac{\sqrt{16-x^4}}{48 x^3}+\frac{1}{48} \int \frac{1}{\sqrt{16-x^4}} \, dx\\ &=-\frac{\sqrt{16-x^4}}{48 x^3}+\frac{1}{96} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0031155, size = 24, normalized size = 0.77 \[ -\frac{\, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{x^4}{16}\right )}{12 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[16 - x^4]),x]

[Out]

-Hypergeometric2F1[-3/4, 1/2, 1/4, x^4/16]/(12*x^3)

________________________________________________________________________________________

Maple [B]  time = 0.009, size = 49, normalized size = 1.6 \begin{align*} -{\frac{1}{48\,{x}^{3}}\sqrt{-{x}^{4}+16}}+{\frac{1}{96}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-x^4+16)^(1/2),x)

[Out]

-1/48*(-x^4+16)^(1/2)/x^3+1/96*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*EllipticF(1/2*x,I)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^4), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 16}}{x^{8} - 16 \, x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + 16)/(x^8 - 16*x^4), x)

________________________________________________________________________________________

Sympy [A]  time = 0.714316, size = 36, normalized size = 1.16 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 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+16)**(1/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4*exp_polar(2*I*pi)/16)/(16*x**3*gamma(1/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^4), x)