3.170 \(\int \frac{x^5}{\left (1+x^4\right )^3} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{16} \tan ^{-1}\left (x^2\right )+\frac{x^2}{16 \left (x^4+1\right )}-\frac{x^2}{8 \left (x^4+1\right )^2} \]

[Out]

-x^2/(8*(1 + x^4)^2) + x^2/(16*(1 + x^4)) + ArcTan[x^2]/16

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Rubi [A]  time = 0.0329375, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{1}{16} \tan ^{-1}\left (x^2\right )+\frac{x^2}{16 \left (x^4+1\right )}-\frac{x^2}{8 \left (x^4+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 + x^4)^3,x]

[Out]

-x^2/(8*(1 + x^4)^2) + x^2/(16*(1 + x^4)) + ArcTan[x^2]/16

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Rubi in Sympy [A]  time = 2.3697, size = 27, normalized size = 0.73 \[ \frac{x^{2}}{16 \left (x^{4} + 1\right )} - \frac{x^{2}}{8 \left (x^{4} + 1\right )^{2}} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**4+1)**3,x)

[Out]

x**2/(16*(x**4 + 1)) - x**2/(8*(x**4 + 1)**2) + atan(x**2)/16

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Mathematica [A]  time = 0.0156113, size = 25, normalized size = 0.68 \[ \frac{1}{16} \left (\tan ^{-1}\left (x^2\right )+\frac{\left (x^4-1\right ) x^2}{\left (x^4+1\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(1 + x^4)^3,x]

[Out]

((x^2*(-1 + x^4))/(1 + x^4)^2 + ArcTan[x^2])/16

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Maple [A]  time = 0.013, size = 28, normalized size = 0.8 \[{\frac{1}{2\, \left ({x}^{4}+1 \right ) ^{2}} \left ({\frac{{x}^{6}}{8}}-{\frac{{x}^{2}}{8}} \right ) }+{\frac{\arctan \left ({x}^{2} \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^4+1)^3,x)

[Out]

1/2*(1/8*x^6-1/8*x^2)/(x^4+1)^2+1/16*arctan(x^2)

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Maxima [A]  time = 1.55799, size = 41, normalized size = 1.11 \[ \frac{x^{6} - x^{2}}{16 \,{\left (x^{8} + 2 \, x^{4} + 1\right )}} + \frac{1}{16} \, \arctan \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^3,x, algorithm="maxima")

[Out]

1/16*(x^6 - x^2)/(x^8 + 2*x^4 + 1) + 1/16*arctan(x^2)

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Fricas [A]  time = 0.221489, size = 51, normalized size = 1.38 \[ \frac{x^{6} - x^{2} +{\left (x^{8} + 2 \, x^{4} + 1\right )} \arctan \left (x^{2}\right )}{16 \,{\left (x^{8} + 2 \, x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^3,x, algorithm="fricas")

[Out]

1/16*(x^6 - x^2 + (x^8 + 2*x^4 + 1)*arctan(x^2))/(x^8 + 2*x^4 + 1)

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Sympy [A]  time = 0.209027, size = 24, normalized size = 0.65 \[ \frac{x^{6} - x^{2}}{16 x^{8} + 32 x^{4} + 16} + \frac{\operatorname{atan}{\left (x^{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**4+1)**3,x)

[Out]

(x**6 - x**2)/(16*x**8 + 32*x**4 + 16) + atan(x**2)/16

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GIAC/XCAS [A]  time = 0.210997, size = 54, normalized size = 1.46 \[ \frac{x^{2} - \frac{1}{x^{2}}}{16 \,{\left ({\left (x^{2} - \frac{1}{x^{2}}\right )}^{2} + 4\right )}} + \frac{1}{32} \, \arctan \left (\frac{x^{4} - 1}{2 \, x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^4 + 1)^3,x, algorithm="giac")

[Out]

1/16*(x^2 - 1/x^2)/((x^2 - 1/x^2)^2 + 4) + 1/32*arctan(1/2*(x^4 - 1)/x^2)