3.347 \(\int \frac{x^m}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=127 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+i\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+i\right ) (m+1)} \]

[Out]

(2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 - I*Sqrt[3])]
)/(Sqrt[3]*(I + Sqrt[3])*(1 + m)) - (2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4,
(5 + m)/4, (2*x^4)/(1 + I*Sqrt[3])])/(Sqrt[3]*(I - Sqrt[3])*(1 + m))

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Rubi [A]  time = 0.114895, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+i\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+i\right ) (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(1 - x^4 + x^8),x]

[Out]

(2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (2*x^4)/(1 - I*Sqrt[3])]
)/(Sqrt[3]*(I + Sqrt[3])*(1 + m)) - (2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4,
(5 + m)/4, (2*x^4)/(1 + I*Sqrt[3])])/(Sqrt[3]*(I - Sqrt[3])*(1 + m))

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Rubi in Sympy [A]  time = 18.287, size = 100, normalized size = 0.79 \[ \frac{2 \sqrt{3} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{x^{4} \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}\right )} \right )}}{3 \left (\sqrt{3} - i\right ) \left (m + 1\right )} + \frac{2 \sqrt{3} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{x^{4} \left (\frac{1}{2} + \frac{\sqrt{3} i}{2}\right )} \right )}}{3 \left (\sqrt{3} + i\right ) \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(x**8-x**4+1),x)

[Out]

2*sqrt(3)*x**(m + 1)*hyper((1, m/4 + 1/4), (m/4 + 5/4,), x**4*(1/2 - sqrt(3)*I/2
))/(3*(sqrt(3) - I)*(m + 1)) + 2*sqrt(3)*x**(m + 1)*hyper((1, m/4 + 1/4), (m/4 +
5/4,), x**4*(1/2 + sqrt(3)*I/2))/(3*(sqrt(3) + I)*(m + 1))

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Mathematica [C]  time = 0.0770746, size = 79, normalized size = 0.62 \[ \frac{x^m \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ]}{4 m} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^m/(1 - x^4 + x^8),x]

[Out]

(x^m*RootSum[1 - #1^4 + #1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))
]/((x/(x - #1))^m*(-#1^3 + 2*#1^7)) & ])/(4*m)

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}-{x}^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(x^8-x^4+1),x)

[Out]

int(x^m/(x^8-x^4+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^m/(x^8 - x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{x^{8} - x^{4} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^m/(x^8 - x^4 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(x**8-x**4+1),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^m/(x^8 - x^4 + 1), x)