Optimal. Leaf size=37 \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0619881, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x^7/(1 + x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 8.2776, size = 34, normalized size = 0.92 \[ \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} + \frac{1}{3}\right ) \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7/(x**8+x**4+1),x)
[Out]
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Mathematica [A] time = 0.0168237, size = 37, normalized size = 1. \[ \frac{1}{8} \log \left (x^8+x^4+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^4+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^7/(1 + x^4 + x^8),x]
[Out]
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Maple [A] time = 0.004, size = 31, normalized size = 0.8 \[{\frac{\ln \left ({x}^{8}+{x}^{4}+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}+1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7/(x^8+x^4+1),x)
[Out]
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Maxima [A] time = 0.821607, size = 41, normalized size = 1.11 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/(x^8 + x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.250371, size = 46, normalized size = 1.24 \[ \frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} + x^{4} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/(x^8 + x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.313318, size = 37, normalized size = 1. \[ \frac{\log{\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7/(x**8+x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.269861, size = 41, normalized size = 1.11 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} + 1\right )}\right ) + \frac{1}{8} \,{\rm ln}\left (x^{8} + x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^7/(x^8 + x^4 + 1),x, algorithm="giac")
[Out]