Optimal. Leaf size=32 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right )}{m+1} \]
[Out]
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Rubi [A] time = 0.0220683, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Int[x^m/(1 + 2*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 4.52388, size = 24, normalized size = 0.75 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- x^{4}} \right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(x**8+2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0201887, size = 34, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+1}{4}+1;-x^4\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/(1 + 2*x^4 + x^8),x]
[Out]
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Maple [F] time = 0.03, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}+2\,{x}^{4}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(x^8+2*x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\left (x^{4} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(x**8+2*x**4+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="giac")
[Out]