Optimal. Leaf size=58 \[ \frac{1}{3} \sqrt{x^4+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+1}} \]
[Out]
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Rubi [A] time = 0.0224954, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{3} \sqrt{x^4+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 1.42838, size = 49, normalized size = 0.84 \[ \frac{x \sqrt{x^{4} + 1}}{3} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0329822, size = 48, normalized size = 0.83 \[ \frac{x^5-2 \sqrt [4]{-1} \sqrt{x^4+1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+x}{3 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + x^4],x]
[Out]
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Maple [C] time = 0.006, size = 72, normalized size = 1.2 \[{\frac{x}{3}\sqrt{{x}^{4}+1}}+{\frac{2\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{3\,\sqrt{2}}{2}}+{\frac{3\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{x^{4} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.69543, size = 29, normalized size = 0.5 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 1),x, algorithm="giac")
[Out]