Optimal. Leaf size=53 \[ \frac{1}{4} \log \left (x^4+1\right )+\frac{1}{2} \tan ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.111879, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{1}{4} \log \left (x^4+1\right )+\frac{1}{2} \tan ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x + x^2 + x^3)/(1 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 16.6124, size = 48, normalized size = 0.91 \[ \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3+x**2+x+1)/(x**4+1),x)
[Out]
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Mathematica [A] time = 0.0642398, size = 50, normalized size = 0.94 \[ \frac{1}{4} \left (\log \left (x^4+1\right )-2 \left (1+\sqrt{2}\right ) \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \left (\sqrt{2}-1\right ) \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x + x^2 + x^3)/(1 + x^4),x]
[Out]
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Maple [B] time = 0.004, size = 102, normalized size = 1.9 \[{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{2}}+{\frac{\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{2}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }+{\frac{\arctan \left ({x}^{2} \right ) }{2}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) }+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3+x^2+x+1)/(x^4+1),x)
[Out]
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Maxima [A] time = 1.51933, size = 103, normalized size = 1.94 \[ -\frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} - 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{4} \, \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - \sqrt{2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 + x^2 + x + 1)/(x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241714, size = 217, normalized size = 4.09 \[ -\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} - 4\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} - 4\right )}}{\left (\sqrt{2} + 1\right )}}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} + 1}\right ) - \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} + 4\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (3 \, \sqrt{2} + 4\right )}}{\left (\sqrt{2} - 1\right )}}{\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} - 1}\right ) + \frac{1}{4} \, \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - \sqrt{2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 + x^2 + x + 1)/(x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.675776, size = 73, normalized size = 1.38 \[ \frac{\log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{4} + \frac{\log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{4} + 2 \left (\frac{1}{4} + \frac{\sqrt{2}}{4}\right ) \operatorname{atan}{\left (\sqrt{2} x - 1 \right )} + 2 \left (- \frac{1}{4} + \frac{\sqrt{2}}{4}\right ) \operatorname{atan}{\left (\sqrt{2} x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3+x**2+x+1)/(x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.211986, size = 95, normalized size = 1.79 \[ \frac{1}{2} \,{\left (\sqrt{2} - 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{2} \,{\left (\sqrt{2} + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 + x^2 + x + 1)/(x^4 + 1),x, algorithm="giac")
[Out]