3.280 \(\int \frac{1}{x^7 \left (1+2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=37 \[ -\frac{5}{12 x^6}+\frac{5}{4 x^2}+\frac{5}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^6 \left (x^4+1\right )} \]

[Out]

-5/(12*x^6) + 5/(4*x^2) + 1/(4*x^6*(1 + x^4)) + (5*ArcTan[x^2])/4

_______________________________________________________________________________________

Rubi [A]  time = 0.0417847, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{5}{12 x^6}+\frac{5}{4 x^2}+\frac{5}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^6 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(1 + 2*x^4 + x^8)),x]

[Out]

-5/(12*x^6) + 5/(4*x^2) + 1/(4*x^6*(1 + x^4)) + (5*ArcTan[x^2])/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.52098, size = 32, normalized size = 0.86 \[ \frac{5 \operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{5}{4 x^{2}} - \frac{5}{12 x^{6}} + \frac{1}{4 x^{6} \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**8+2*x**4+1),x)

[Out]

5*atan(x**2)/4 + 5/(4*x**2) - 5/(12*x**6) + 1/(4*x**6*(x**4 + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0163911, size = 33, normalized size = 0.89 \[ -\frac{1}{6 x^6}+\frac{1}{x^2}-\frac{5}{4} \tan ^{-1}\left (\frac{1}{x^2}\right )+\frac{x^2}{4 \left (x^4+1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(1 + 2*x^4 + x^8)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 28, normalized size = 0.8 \[{\frac{{x}^{2}}{4\,{x}^{4}+4}}+{\frac{5\,\arctan \left ({x}^{2} \right ) }{4}}-{\frac{1}{6\,{x}^{6}}}+{x}^{-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^8+2*x^4+1),x)

[Out]

1/4*x^2/(x^4+1)+5/4*arctan(x^2)-1/6/x^6+1/x^2

_______________________________________________________________________________________

Maxima [A]  time = 0.860928, size = 41, normalized size = 1.11 \[ \frac{15 \, x^{8} + 10 \, x^{4} - 2}{12 \,{\left (x^{10} + x^{6}\right )}} + \frac{5}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^7),x, algorithm="maxima")

[Out]

1/12*(15*x^8 + 10*x^4 - 2)/(x^10 + x^6) + 5/4*arctan(x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.245721, size = 49, normalized size = 1.32 \[ \frac{15 \, x^{8} + 10 \, x^{4} + 15 \,{\left (x^{10} + x^{6}\right )} \arctan \left (x^{2}\right ) - 2}{12 \,{\left (x^{10} + x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^7),x, algorithm="fricas")

[Out]

1/12*(15*x^8 + 10*x^4 + 15*(x^10 + x^6)*arctan(x^2) - 2)/(x^10 + x^6)

_______________________________________________________________________________________

Sympy [A]  time = 0.515027, size = 29, normalized size = 0.78 \[ \frac{5 \operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{15 x^{8} + 10 x^{4} - 2}{12 x^{10} + 12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**8+2*x**4+1),x)

[Out]

5*atan(x**2)/4 + (15*x**8 + 10*x**4 - 2)/(12*x**10 + 12*x**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271181, size = 42, normalized size = 1.14 \[ \frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} + \frac{6 \, x^{4} - 1}{6 \, x^{6}} + \frac{5}{4} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x^7),x, algorithm="giac")

[Out]

1/4*x^2/(x^4 + 1) + 1/6*(6*x^4 - 1)/x^6 + 5/4*arctan(x^2)