3.117 \(\int \frac{1+x^2}{\left (16+x^2\right )^3} \, dx\)

Optimal. Leaf size=35 \[ \frac{19 x}{2048 \left (x^2+16\right )}-\frac{15 x}{64 \left (x^2+16\right )^2}+\frac{19 \tan ^{-1}\left (\frac{x}{4}\right )}{8192} \]

[Out]

(-15*x)/(64*(16 + x^2)^2) + (19*x)/(2048*(16 + x^2)) + (19*ArcTan[x/4])/8192

_______________________________________________________________________________________

Rubi [A]  time = 0.0234301, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{19 x}{2048 \left (x^2+16\right )}-\frac{15 x}{64 \left (x^2+16\right )^2}+\frac{19 \tan ^{-1}\left (\frac{x}{4}\right )}{8192} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(16 + x^2)^3,x]

[Out]

(-15*x)/(64*(16 + x^2)^2) + (19*x)/(2048*(16 + x^2)) + (19*ArcTan[x/4])/8192

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.13407, size = 29, normalized size = 0.83 \[ \frac{19 x}{2048 \left (x^{2} + 16\right )} - \frac{15 x}{64 \left (x^{2} + 16\right )^{2}} + \frac{19 \operatorname{atan}{\left (\frac{x}{4} \right )}}{8192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)/(x**2+16)**3,x)

[Out]

19*x/(2048*(x**2 + 16)) - 15*x/(64*(x**2 + 16)**2) + 19*atan(x/4)/8192

_______________________________________________________________________________________

Mathematica [A]  time = 0.0153569, size = 35, normalized size = 1. \[ \frac{19 x}{2048 \left (x^2+16\right )}-\frac{15 x}{64 \left (x^2+16\right )^2}+\frac{19 \tan ^{-1}\left (\frac{x}{4}\right )}{8192} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)/(16 + x^2)^3,x]

[Out]

(-15*x)/(64*(16 + x^2)^2) + (19*x)/(2048*(16 + x^2)) + (19*ArcTan[x/4])/8192

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 25, normalized size = 0.7 \[{\frac{1}{ \left ({x}^{2}+16 \right ) ^{2}} \left ({\frac{19\,{x}^{3}}{2048}}-{\frac{11\,x}{128}} \right ) }+{\frac{19}{8192}\arctan \left ({\frac{x}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^2+16)^3,x)

[Out]

(19/2048*x^3-11/128*x)/(x^2+16)^2+19/8192*arctan(1/4*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.50787, size = 41, normalized size = 1.17 \[ \frac{19 \, x^{3} - 176 \, x}{2048 \,{\left (x^{4} + 32 \, x^{2} + 256\right )}} + \frac{19}{8192} \, \arctan \left (\frac{1}{4} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^2 + 16)^3,x, algorithm="maxima")

[Out]

1/2048*(19*x^3 - 176*x)/(x^4 + 32*x^2 + 256) + 19/8192*arctan(1/4*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.24074, size = 53, normalized size = 1.51 \[ \frac{76 \, x^{3} + 19 \,{\left (x^{4} + 32 \, x^{2} + 256\right )} \arctan \left (\frac{1}{4} \, x\right ) - 704 \, x}{8192 \,{\left (x^{4} + 32 \, x^{2} + 256\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^2 + 16)^3,x, algorithm="fricas")

[Out]

1/8192*(76*x^3 + 19*(x^4 + 32*x^2 + 256)*arctan(1/4*x) - 704*x)/(x^4 + 32*x^2 +
256)

_______________________________________________________________________________________

Sympy [A]  time = 0.311551, size = 27, normalized size = 0.77 \[ \frac{19 x^{3} - 176 x}{2048 x^{4} + 65536 x^{2} + 524288} + \frac{19 \operatorname{atan}{\left (\frac{x}{4} \right )}}{8192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)/(x**2+16)**3,x)

[Out]

(19*x**3 - 176*x)/(2048*x**4 + 65536*x**2 + 524288) + 19*atan(x/4)/8192

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227144, size = 34, normalized size = 0.97 \[ \frac{19 \, x^{3} - 176 \, x}{2048 \,{\left (x^{2} + 16\right )}^{2}} + \frac{19}{8192} \, \arctan \left (\frac{1}{4} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/(x^2 + 16)^3,x, algorithm="giac")

[Out]

1/2048*(19*x^3 - 176*x)/(x^2 + 16)^2 + 19/8192*arctan(1/4*x)