3.1.34 \(\int \frac {1}{1+2 x^2+x^4} \, dx\) [34]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 205, 209} \begin {gather*} \frac {\arctan (x)}{2}+\frac {x}{2 \left (x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2 + x^4)^(-1),x]

[Out]

x/(2*(1 + x^2)) + ArcTan[x]/2

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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\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 {\arctan (x)}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{2} \left (\frac {x}{1+x^2}+\arctan (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2 + x^4)^(-1),x]

[Out]

(x/(1 + x^2) + ArcTan[x])/2

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Maple [A]
time = 0.02, size = 16, normalized size = 0.84

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.43, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)

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Fricas [A]
time = 0.56, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.05, size = 12, normalized size = 0.63 \begin {gather*} \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(2*x**2 + 2) + atan(x)/2

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Giac [A]
time = 0.46, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/(x^2 + 1) + 1/2*arctan(x)

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Mupad [B]
time = 0.06, size = 16, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{2}+\frac {x}{2\,\left (x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2 + x^4 + 1),x)

[Out]

atan(x)/2 + x/(2*(x^2 + 1))

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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.58 \begin {gather*} -\frac {1}{2 x^{2}+2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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