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]
________________________________________________________________________________________
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]
[Out]
Rule 325
Rule 221
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]