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

Optimal. Leaf size=35 \[ \frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

-(ArcTan[1 - Sqrt[2]*x]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.0183083, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1162, 617, 204} \[ \frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + x^4),x]

[Out]

-(ArcTan[1 - Sqrt[2]*x]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*x]/Sqrt[2]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{1+x^2}{1+x^4} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{\sqrt{2}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (1+\sqrt{2} x\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0129433, size = 30, normalized size = 0.86 \[ \frac{\tan ^{-1}\left (\sqrt{2} x+1\right )-\tan ^{-1}\left (1-\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(1 + x^4),x]

[Out]

(-ArcTan[1 - Sqrt[2]*x] + ArcTan[1 + Sqrt[2]*x])/Sqrt[2]

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Maple [B]  time = 0.044, size = 88, normalized size = 2.5 \begin{align*}{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{2}}+{\frac{\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{2}}+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }+{\frac{\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(1+x*2^(1/2))*2^(1/2)+1/2*arctan(-1+x*2^(1/2))*2^(1/2)+1/8*2^(1/2)*ln((1+x^2+x*2^(1/2))/(1+x^2-x*2^(
1/2)))+1/8*2^(1/2)*ln((1+x^2-x*2^(1/2))/(1+x^2+x*2^(1/2)))

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Maxima [A]  time = 1.45257, size = 53, normalized size = 1.51 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)))

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Fricas [A]  time = 1.37549, size = 107, normalized size = 3.06 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{3} + x\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(x^3 + x)) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*x)

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Sympy [A]  time = 0.101463, size = 39, normalized size = 1.11 \begin{align*} \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{2} x^{3}}{2} + \frac{\sqrt{2} x}{2} \right )}\right )}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(2*atan(sqrt(2)*x/2) + 2*atan(sqrt(2)*x**3/2 + sqrt(2)*x/2))/4

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Giac [A]  time = 1.13747, size = 53, normalized size = 1.51 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)))