\(\int \frac {3+2 x^2}{(1+x^2)^2} \, dx\) [111]

Optimal result

Integrand size = 15, antiderivative size = 19 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{2 \left (1+x^2\right )}+\frac {5 \arctan (x)}{2} \]

[Out]

1/2*x/(x^2+1)+5/2*arctan(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {393, 209} \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {5 \arctan (x)}{2}+\frac {x}{2 \left (x^2+1\right )} \]

[In]

Int[(3 + 2*x^2)/(1 + x^2)^2,x]

[Out]

x/(2*(1 + x^2)) + (5*ArcTan[x])/2

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {x}{2 \left (1+x^2\right )}+\frac {5}{2} \int \frac {1}{1+x^2} \, dx \\ & = \frac {x}{2 \left (1+x^2\right )}+\frac {5}{2} \tan ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{2 \left (1+x^2\right )}+\frac {5 \arctan (x)}{2} \]

[In]

Integrate[(3 + 2*x^2)/(1 + x^2)^2,x]

[Out]

x/(2*(1 + x^2)) + (5*ArcTan[x])/2

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {x}{2 x^{2}+2}+\frac {5 \arctan \left (x \right )}{2}\)	\(16\)
risch	\(\frac {x}{2 x^{2}+2}+\frac {5 \arctan \left (x \right )}{2}\)	\(16\)
meijerg	\(-\frac {x}{x^{2}+1}+\frac {5 \arctan \left (x \right )}{2}+\frac {3 x}{2 x^{2}+2}\)	\(28\)
parallelrisch	\(-\frac {5 i \ln \left (x -i\right ) x^{2}-5 i \ln \left (x +i\right ) x^{2}+5 i \ln \left (x -i\right )-5 i \ln \left (x +i\right )-2 x}{4 \left (x^{2}+1\right )}\)	\(52\)

[In]

int((2*x^2+3)/(x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*x/(x^2+1)+5/2*arctan(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {5 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \, {\left (x^{2} + 1\right )}} \]

[In]

integrate((2*x^2+3)/(x^2+1)^2,x, algorithm="fricas")

[Out]

1/2*(5*(x^2 + 1)*arctan(x) + x)/(x^2 + 1)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{2 x^{2} + 2} + \frac {5 \operatorname {atan}{\left (x \right )}}{2} \]

[In]

integrate((2*x**2+3)/(x**2+1)**2,x)

[Out]

x/(2*x**2 + 2) + 5*atan(x)/2

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {5}{2} \, \arctan \left (x\right ) \]

[In]

integrate((2*x^2+3)/(x^2+1)^2,x, algorithm="maxima")

[Out]

1/2*x/(x^2 + 1) + 5/2*arctan(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {5}{2} \, \arctan \left (x\right ) \]

[In]

integrate((2*x^2+3)/(x^2+1)^2,x, algorithm="giac")

[Out]

1/2*x/(x^2 + 1) + 5/2*arctan(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {3+2 x^2}{\left (1+x^2\right )^2} \, dx=\frac {5\,\mathrm {atan}\left (x\right )}{2}+\frac {x}{2\,\left (x^2+1\right )} \]

[In]

int((2*x^2 + 3)/(x^2 + 1)^2,x)

[Out]

(5*atan(x))/2 + x/(2*(x^2 + 1))