3.161 \(\int \frac {1}{(1+x^2)^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \tan ^{-1}(x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {199, 203} \[ \frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)^(-2),x]

[Out]

x/(2*(1 + x^2)) + ArcTan[x]/2

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 16, normalized size = 0.84 \[ \frac {1}{2} \left (\frac {x}{x^2+1}+\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)^(-2),x]

[Out]

(x/(1 + x^2) + ArcTan[x])/2

________________________________________________________________________________________

fricas [A]  time = 0.37, size = 19, normalized size = 1.00 \[ \frac {{\left (x^{2} + 1\right )} \arctan \relax (x) + x}{2 \, {\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.08, size = 15, normalized size = 0.79 \[ \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ \frac {x}{2 x^{2}+2}+\frac {\arctan \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.16, size = 15, normalized size = 0.79 \[ \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 16, normalized size = 0.84 \[ \frac {\mathrm {atan}\relax (x)}{2}+\frac {x}{2\,\left (x^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 1)^2,x)

[Out]

atan(x)/2 + x/(2*(x^2 + 1))

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 12, normalized size = 0.63 \[ \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(2*x**2 + 2) + atan(x)/2

________________________________________________________________________________________