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

Optimal. Leaf size=26 \[ \frac{x^3}{3}-2 x+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-2*x + x^3/3 + (5*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Rubi [A]  time = 0.0107413, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1154, 203} \[ \frac{x^3}{3}-2 x+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(2 + x^2),x]

[Out]

-2*x + x^3/3 + (5*ArcTan[x/Sqrt[2]])/Sqrt[2]

Rule 1154

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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^4}{2+x^2} \, dx &=\int \left (-2+x^2+\frac{5}{2+x^2}\right ) \, dx\\ &=-2 x+\frac{x^3}{3}+5 \int \frac{1}{2+x^2} \, dx\\ &=-2 x+\frac{x^3}{3}+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0080328, size = 26, normalized size = 1. \[ \frac{x^3}{3}-2 x+\frac{5 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(2 + x^2),x]

[Out]

-2*x + x^3/3 + (5*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Maple [A]  time = 0.003, size = 22, normalized size = 0.9 \begin{align*} -2\,x+{\frac{{x}^{3}}{3}}+{\frac{5\,\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x+1/3*x^3+5/2*arctan(1/2*x*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.4592, size = 28, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} + \frac{5}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 5/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*x

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Fricas [A]  time = 1.63274, size = 69, normalized size = 2.65 \begin{align*} \frac{1}{3} \, x^{3} + \frac{5}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 5/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*x

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Sympy [A]  time = 0.086571, size = 26, normalized size = 1. \begin{align*} \frac{x^{3}}{3} - 2 x + \frac{5 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 - 2*x + 5*sqrt(2)*atan(sqrt(2)*x/2)/2

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Giac [A]  time = 1.22294, size = 28, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} + \frac{5}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 5/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*x