3.1360 \(\int \frac{x^7}{1+x^6} \, dx\)

Optimal. Leaf size=56 \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

[Out]

x^2/2 + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 + x^2]/6 + Log[1 - x^2 +
x^4]/12

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Rubi [A]  time = 0.0853177, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ \frac{x^2}{2}-\frac{1}{6} \log \left (x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(1 + x^6),x]

[Out]

x^2/2 + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 + x^2]/6 + Log[1 - x^2 +
x^4]/12

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Rubi in Sympy [A]  time = 10.3503, size = 48, normalized size = 0.86 \[ \frac{x^{2}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} - \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**6+1),x)

[Out]

x**2/2 - log(x**2 + 1)/6 + log(x**4 - x**2 + 1)/12 - sqrt(3)*atan(sqrt(3)*(2*x**
2/3 - 1/3))/6

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Mathematica [A]  time = 0.0297741, size = 79, normalized size = 1.41 \[ \frac{1}{12} \left (6 x^2-2 \log \left (x^2+1\right )+\log \left (x^2-\sqrt{3} x+1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(1 + x^6),x]

[Out]

(6*x^2 + 2*Sqrt[3]*ArcTan[Sqrt[3] - 2*x] + 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*x] - 2*L
og[1 + x^2] + Log[1 - Sqrt[3]*x + x^2] + Log[1 + Sqrt[3]*x + x^2])/12

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Maple [A]  time = 0.01, size = 46, normalized size = 0.8 \[{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^6+1),x)

[Out]

1/2*x^2+1/12*ln(x^4-x^2+1)-1/6*3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))-1/6*ln(x^2+
1)

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Maxima [A]  time = 1.58688, size = 61, normalized size = 1.09 \[ \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 1),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) + 1/12*log(x^4 - x^2 + 1)
- 1/6*log(x^2 + 1)

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Fricas [A]  time = 0.21937, size = 74, normalized size = 1.32 \[ \frac{1}{36} \, \sqrt{3}{\left (6 \, \sqrt{3} x^{2} + \sqrt{3} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, \sqrt{3} \log \left (x^{2} + 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 1),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(6*sqrt(3)*x^2 + sqrt(3)*log(x^4 - x^2 + 1) - 2*sqrt(3)*log(x^2 + 1
) - 6*arctan(1/3*sqrt(3)*(2*x^2 - 1)))

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Sympy [A]  time = 0.418709, size = 51, normalized size = 0.91 \[ \frac{x^{2}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**6+1),x)

[Out]

x**2/2 - log(x**2 + 1)/6 + log(x**4 - x**2 + 1)/12 - sqrt(3)*atan(2*sqrt(3)*x**2
/3 - sqrt(3)/3)/6

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GIAC/XCAS [A]  time = 0.221672, size = 61, normalized size = 1.09 \[ \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 1),x, algorithm="giac")

[Out]

1/2*x^2 - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) + 1/12*ln(x^4 - x^2 + 1) -
1/6*ln(x^2 + 1)