3.322 \(\int \frac{9+x^4}{x^2 \left (9+x^2\right )} \, dx\)

Optimal. Leaf size=17 \[ x-\frac{1}{x}-\frac{10}{3} \tan ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

-x^(-1) + x - (10*ArcTan[x/3])/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0415431, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ x-\frac{1}{x}-\frac{10}{3} \tan ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(9 + x^4)/(x^2*(9 + x^2)),x]

[Out]

-x^(-1) + x - (10*ArcTan[x/3])/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.98288, size = 12, normalized size = 0.71 \[ x - \frac{10 \operatorname{atan}{\left (\frac{x}{3} \right )}}{3} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+9)/x**2/(x**2+9),x)

[Out]

x - 10*atan(x/3)/3 - 1/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.00869106, size = 17, normalized size = 1. \[ x-\frac{1}{x}-\frac{10}{3} \tan ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(9 + x^4)/(x^2*(9 + x^2)),x]

[Out]

-x^(-1) + x - (10*ArcTan[x/3])/3

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 14, normalized size = 0.8 \[ -{x}^{-1}+x-{\frac{10}{3}\arctan \left ({\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+9)/x^2/(x^2+9),x)

[Out]

-1/x+x-10/3*arctan(1/3*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.88416, size = 18, normalized size = 1.06 \[ x - \frac{1}{x} - \frac{10}{3} \, \arctan \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 9)/((x^2 + 9)*x^2),x, algorithm="maxima")

[Out]

x - 1/x - 10/3*arctan(1/3*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.252626, size = 26, normalized size = 1.53 \[ \frac{3 \, x^{2} - 10 \, x \arctan \left (\frac{1}{3} \, x\right ) - 3}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 9)/((x^2 + 9)*x^2),x, algorithm="fricas")

[Out]

1/3*(3*x^2 - 10*x*arctan(1/3*x) - 3)/x

_______________________________________________________________________________________

Sympy [A]  time = 0.208994, size = 12, normalized size = 0.71 \[ x - \frac{10 \operatorname{atan}{\left (\frac{x}{3} \right )}}{3} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+9)/x**2/(x**2+9),x)

[Out]

x - 10*atan(x/3)/3 - 1/x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.259991, size = 18, normalized size = 1.06 \[ x - \frac{1}{x} - \frac{10}{3} \, \arctan \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 9)/((x^2 + 9)*x^2),x, algorithm="giac")

[Out]

x - 1/x - 10/3*arctan(1/3*x)