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

Optimal. Leaf size=20 \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

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Rubi [A]  time = 0.0121168, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {522, 203} \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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

Mathematica [A]  time = 0.0089066, size = 20, normalized size = 1. \[ 3 \tan ^{-1}(x)-\sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*ArcTan[x] - Sqrt[2]*ArcTan[x/Sqrt[2]]

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Maple [A]  time = 0.006, size = 18, normalized size = 0.9 \begin{align*} 3\,\arctan \left ( x \right ) -\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.54356, size = 23, normalized size = 1.15 \begin{align*} -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67032, size = 62, normalized size = 3.1 \begin{align*} -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.132992, size = 19, normalized size = 0.95 \begin{align*} 3 \operatorname{atan}{\left (x \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*atan(x) - sqrt(2)*atan(sqrt(2)*x/2)

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Giac [A]  time = 1.0889, size = 23, normalized size = 1.15 \begin{align*} -\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 3 \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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