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]
________________________________________________________________________________________
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]
[Out]
Rule 325
Rule 222
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]