3.160 \(\int \frac{x^7}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=119 \[ \frac{x^2}{2}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{1}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\frac{1}{6} \log (x+1)+\frac{1}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{3}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

[Out]

x^2/2 - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + (3*3^(1/6)*ArcTan[(3^(1/3) - 2*x
)/3^(5/6)])/2 - Log[1 + x]/6 + (3^(2/3)*Log[3^(1/3) + x])/2 + Log[1 - x + x^2]/1
2 - (3^(2/3)*Log[3^(2/3) - 3^(1/3)*x + x^2])/4

_______________________________________________________________________________________

Rubi [A]  time = 0.167437, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ \frac{x^2}{2}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{1}{4} 3^{2/3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\frac{1}{6} \log (x+1)+\frac{1}{2} 3^{2/3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{3}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(3 + 4*x^3 + x^6),x]

[Out]

x^2/2 - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + (3*3^(1/6)*ArcTan[(3^(1/3) - 2*x
)/3^(5/6)])/2 - Log[1 + x]/6 + (3^(2/3)*Log[3^(1/3) + x])/2 + Log[1 - x + x^2]/1
2 - (3^(2/3)*Log[3^(2/3) - 3^(1/3)*x + x^2])/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.7032, size = 110, normalized size = 0.92 \[ \frac{x^{2}}{2} - \frac{\log{\left (x + 1 \right )}}{6} + \frac{3^{\frac{2}{3}} \log{\left (x + \sqrt [3]{3} \right )}}{2} + \frac{\log{\left (x^{2} - x + 1 \right )}}{12} - \frac{3^{\frac{2}{3}} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{3 \sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**6+4*x**3+3),x)

[Out]

x**2/2 - log(x + 1)/6 + 3**(2/3)*log(x + 3**(1/3))/2 + log(x**2 - x + 1)/12 - 3*
*(2/3)*log(x**2 - 3**(1/3)*x + 3**(2/3))/4 + sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))
/6 + 3*3**(1/6)*atan(sqrt(3)*(-2*3**(2/3)*x/9 + 1/3))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0484352, size = 111, normalized size = 0.93 \[ \frac{1}{12} \left (6 x^2+\log \left (x^2-x+1\right )-3\ 3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-2 \log (x+1)+6\ 3^{2/3} \log \left (3^{2/3} x+3\right )+18 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(3 + 4*x^3 + x^6),x]

[Out]

(6*x^2 + 18*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x
)/Sqrt[3]] - 2*Log[1 + x] + 6*3^(2/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3*
3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 89, normalized size = 0.8 \[{\frac{{x}^{2}}{2}}+{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{2}}-{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{4}}-{\frac{3\,\sqrt [6]{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^6+4*x^3+3),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.867801, size = 120, normalized size = 1.01 \[ \frac{1}{2} \, x^{2} - \frac{1}{4} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{2} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{3}{2} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.278836, size = 154, normalized size = 1.29 \[ \frac{1}{36} \, \sqrt{3}{\left (6 \, \sqrt{3} x^{2} - 3 \cdot 9^{\frac{1}{3}} \sqrt{3} \log \left (3 \, x^{2} - 9^{\frac{2}{3}} x + 3 \cdot 9^{\frac{1}{3}}\right ) + 6 \cdot 9^{\frac{1}{3}} \sqrt{3} \log \left (3 \, x + 9^{\frac{2}{3}}\right ) + \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \log \left (x + 1\right ) + 18 \cdot 9^{\frac{1}{3}} \arctan \left (-\frac{1}{27} \cdot 9^{\frac{1}{3}} \sqrt{3}{\left (6 \, x - 9^{\frac{2}{3}}\right )}\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(6*sqrt(3)*x^2 - 3*9^(1/3)*sqrt(3)*log(3*x^2 - 9^(2/3)*x + 3*9^(1/3
)) + 6*9^(1/3)*sqrt(3)*log(3*x + 9^(2/3)) + sqrt(3)*log(x^2 - x + 1) - 2*sqrt(3)
*log(x + 1) + 18*9^(1/3)*arctan(-1/27*9^(1/3)*sqrt(3)*(6*x - 9^(2/3))) + 6*arcta
n(1/3*sqrt(3)*(2*x - 1)))

_______________________________________________________________________________________

Sympy [A]  time = 2.24524, size = 134, normalized size = 1.13 \[ \frac{x^{2}}{2} - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{6562 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2}}{183} - \frac{1872 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5}}{61} \right )} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{1872 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5}}{61} + \frac{6562 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2}}{183} \right )} + \operatorname{RootSum}{\left (8 t^{3} - 9, \left ( t \mapsto t \log{\left (- \frac{1872 t^{5}}{61} + \frac{6562 t^{2}}{183} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**6+4*x**3+3),x)

[Out]

x**2/2 - log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 6562*(1/12 - sqrt(3)*I/12)
**2/183 - 1872*(1/12 - sqrt(3)*I/12)**5/61) + (1/12 + sqrt(3)*I/12)*log(x - 1872
*(1/12 + sqrt(3)*I/12)**5/61 + 6562*(1/12 + sqrt(3)*I/12)**2/183) + RootSum(8*_t
**3 - 9, Lambda(_t, _t*log(-1872*_t**5/61 + 6562*_t**2/183 + x)))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError