Optimal. Leaf size=97 \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0930661, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x^4 + x^8)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 14.3421, size = 88, normalized size = 0.91 \[ \frac{x}{4 \left (x^{4} + 1\right )} - \frac{3 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**8+2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0848057, size = 91, normalized size = 0.94 \[ \frac{1}{32} \left (\frac{8 x}{x^4+1}-3 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+3 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x^4 + x^8)^(-1),x]
[Out]
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Maple [A] time = 0.005, size = 68, normalized size = 0.7 \[{\frac{x}{4\,{x}^{4}+4}}+{\frac{3\,\arctan \left ( 1+\sqrt{2}x \right ) \sqrt{2}}{16}}+{\frac{3\,\arctan \left ( \sqrt{2}x-1 \right ) \sqrt{2}}{16}}+{\frac{3\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+\sqrt{2}x}{1+{x}^{2}-\sqrt{2}x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^8+2*x^4+1),x)
[Out]
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Maxima [A] time = 0.86165, size = 111, normalized size = 1.14 \[ \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + 2*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281308, size = 174, normalized size = 1.79 \[ -\frac{12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) + 12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) - 3 \, \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 3 \, \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 8 \, x}{32 \,{\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + 2*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.511355, size = 88, normalized size = 0.91 \[ \frac{x}{4 x^{4} + 4} - \frac{3 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**8+2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.276658, size = 111, normalized size = 1.14 \[ \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2}{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{32} \, \sqrt{2}{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^8 + 2*x^4 + 1),x, algorithm="giac")
[Out]