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

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

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, 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} \begin {gather*} \tan ^{-1}(x) \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \tan ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{1+2 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.10, size = 2, normalized size = 1.00 \begin {gather*} \arctan \relax (x) \end {gather*}

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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giac [A]  time = 0.16, size = 2, normalized size = 1.00 \begin {gather*} \arctan \relax (x) \end {gather*}

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)

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maple [A]  time = 0.00, size = 3, normalized size = 1.50 \begin {gather*} \arctan \relax (x ) \end {gather*}

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 = 2.42, size = 2, normalized size = 1.00 \begin {gather*} \arctan \relax (x) \end {gather*}

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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mupad [B]  time = 4.33, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {atan}\relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/(2*x^2 + x^4 + 1),x)

[Out]

atan(x)

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sympy [A]  time = 0.10, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {atan}{\relax (x )} \end {gather*}

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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