3.72 \(\int \frac{1+x^2}{1+2 x^2+x^4} \, dx\)

Optimal. Leaf size=2 \[ \tan ^{-1}(x) \]

[Out]

ArcTan[x]

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Rubi [A]  time = 0.001515, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {28, 203} \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + 2*x^2 + x^4),x]

[Out]

ArcTan[x]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[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+x^2}{1+2 x^2+x^4} \, dx &=\int \frac{1}{1+x^2} \, dx\\ &=\tan ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0025709, size = 2, normalized size = 1. \[ \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(1 + 2*x^2 + x^4),x]

[Out]

ArcTan[x]

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Maple [A]  time = 0.044, size = 3, normalized size = 1.5 \begin{align*} \arctan \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^4+2*x^2+1),x)

[Out]

arctan(x)

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Maxima [A]  time = 1.43537, size = 3, normalized size = 1.5 \begin{align*} \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+2*x^2+1),x, algorithm="maxima")

[Out]

arctan(x)

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Fricas [A]  time = 1.33545, size = 15, normalized size = 7.5 \begin{align*} \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+2*x^2+1),x, algorithm="fricas")

[Out]

arctan(x)

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Sympy [A]  time = 0.083135, size = 2, normalized size = 1. \begin{align*} \operatorname{atan}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(x**4+2*x**2+1),x)

[Out]

atan(x)

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Giac [A]  time = 1.10475, size = 3, normalized size = 1.5 \begin{align*} \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^4+2*x^2+1),x, algorithm="giac")

[Out]

arctan(x)