Optimal. Leaf size=27 \[ \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-\frac{\sqrt{1-x^4}}{x}-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.0157373, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {325, 307, 221, 1181, 424} \[ -\frac{\sqrt{1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 325
Rule 307
Rule 221
Rule 1181
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{1-x^4}} \, dx &=-\frac{\sqrt{1-x^4}}{x}-\int \frac{x^2}{\sqrt{1-x^4}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{x}+\int \frac{1}{\sqrt{1-x^4}} \, dx-\int \frac{1+x^2}{\sqrt{1-x^4}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx\\ &=-\frac{\sqrt{1-x^4}}{x}-E\left (\left .\sin ^{-1}(x)\right |-1\right )+F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0026997, size = 18, normalized size = 0.67 \[ -\frac{\, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};x^4\right )}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 53, normalized size = 2. \begin{align*} -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+{({\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 1}}{x^{6} - x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.810166, size = 32, normalized size = 1.19 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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