3.31 \(\int \frac{x^7}{1+x^{12}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{12} \log \left (x^4+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{24} \log \left (x^8-x^4+1\right ) \]

[Out]

-ArcTan[(1 - 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4]/12 + Log[1 - x^4 + x^8]/
24

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Rubi [A]  time = 0.0761665, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{1}{12} \log \left (x^4+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{24} \log \left (x^8-x^4+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 + x^12),x]

[Out]

-ArcTan[(1 - 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) - Log[1 + x^4]/12 + Log[1 - x^4 + x^8]/
24

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Rubi in Sympy [A]  time = 3.95518, size = 42, normalized size = 0.86 \[ - \frac{\log{\left (x^{4} + 1 \right )}}{12} + \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{24} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**12+1),x)

[Out]

-log(x**4 + 1)/12 + log(x**8 - x**4 + 1)/24 + sqrt(3)*atan(sqrt(3)*(2*x**4/3 - 1
/3))/12

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Mathematica [B]  time = 0.198422, size = 260, normalized size = 5.31 \[ \frac{1}{24} \left (-2 \log \left (x^2-\sqrt{2} x+1\right )-2 \log \left (x^2+\sqrt{2} x+1\right )+\log \left (2 x^2-\sqrt{6} x+\sqrt{2} x+2\right )+\log \left (2 x^2+\sqrt{2} \left (\sqrt{3}-1\right ) x+2\right )+\log \left (2 x^2-\left (\sqrt{2}+\sqrt{6}\right ) x+2\right )+\log \left (2 x^2+\left (\sqrt{2}+\sqrt{6}\right ) x+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{-2 \sqrt{2} x+\sqrt{3}+1}{1-\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt{2} x-\sqrt{3}+1}{1+\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt{2} x+\sqrt{3}-1}{1+\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt{2} x+\sqrt{3}+1}{\sqrt{3}-1}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 + x^12),x]

[Out]

(2*Sqrt[3]*ArcTan[(1 + Sqrt[3] - 2*Sqrt[2]*x)/(1 - Sqrt[3])] - 2*Sqrt[3]*ArcTan[
(1 - Sqrt[3] + 2*Sqrt[2]*x)/(1 + Sqrt[3])] + 2*Sqrt[3]*ArcTan[(-1 + Sqrt[3] + 2*
Sqrt[2]*x)/(1 + Sqrt[3])] - 2*Sqrt[3]*ArcTan[(1 + Sqrt[3] + 2*Sqrt[2]*x)/(-1 + S
qrt[3])] - 2*Log[1 - Sqrt[2]*x + x^2] - 2*Log[1 + Sqrt[2]*x + x^2] + Log[2 + Sqr
t[2]*x - Sqrt[6]*x + 2*x^2] + Log[2 + Sqrt[2]*(-1 + Sqrt[3])*x + 2*x^2] + Log[2
- (Sqrt[2] + Sqrt[6])*x + 2*x^2] + Log[2 + (Sqrt[2] + Sqrt[6])*x + 2*x^2])/24

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Maple [A]  time = 0.001, size = 41, normalized size = 0.8 \[ -{\frac{\ln \left ({x}^{4}+1 \right ) }{12}}+{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{24}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^12+1),x)

[Out]

-1/12*ln(x^4+1)+1/24*ln(x^8-x^4+1)+1/12*3^(1/2)*arctan(1/3*(2*x^4-1)*3^(1/2))

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Maxima [A]  time = 1.53504, size = 54, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + \frac{1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac{1}{12} \, \log \left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^12 + 1),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 1/24*log(x^8 - x^4 + 1) - 1/12*lo
g(x^4 + 1)

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Fricas [A]  time = 0.209945, size = 63, normalized size = 1.29 \[ \frac{1}{72} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} - x^{4} + 1\right ) - 2 \, \sqrt{3} \log \left (x^{4} + 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^12 + 1),x, algorithm="fricas")

[Out]

1/72*sqrt(3)*(sqrt(3)*log(x^8 - x^4 + 1) - 2*sqrt(3)*log(x^4 + 1) + 6*arctan(1/3
*sqrt(3)*(2*x^4 - 1)))

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Sympy [A]  time = 0.258389, size = 46, normalized size = 0.94 \[ - \frac{\log{\left (x^{4} + 1 \right )}}{12} + \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{24} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**12+1),x)

[Out]

-log(x**4 + 1)/12 + log(x**8 - x**4 + 1)/24 + sqrt(3)*atan(2*sqrt(3)*x**4/3 - sq
rt(3)/3)/12

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GIAC/XCAS [A]  time = 0.208241, size = 54, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) + \frac{1}{24} \,{\rm ln}\left (x^{8} - x^{4} + 1\right ) - \frac{1}{12} \,{\rm ln}\left (x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^12 + 1),x, algorithm="giac")

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) + 1/24*ln(x^8 - x^4 + 1) - 1/12*ln(
x^4 + 1)