Optimal. Leaf size=43 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0030825, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {220} \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+x^4}} \, dx &=\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.011461, size = 21, normalized size = 0.49 \[ -\sqrt [4]{-1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.065, size = 60, normalized size = 1.4 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\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}}\,{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{1}{\sqrt{x^{4} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.657707, size = 27, normalized size = 0.63 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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