### 3.169 $$\int \frac{2-4 x^2+x^3}{(1+x^2) (2+x^2)} \, dx$$

Optimal. Leaf size=36 $-\frac{1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

[Out]

6*ArcTan[x] - 5*Sqrt[2]*ArcTan[x/Sqrt[2]] - Log[1 + x^2]/2 + Log[2 + x^2]

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Rubi [A]  time = 0.117578, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.16, Rules used = {6725, 635, 203, 260} $-\frac{1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 - 4*x^2 + x^3)/((1 + x^2)*(2 + x^2)),x]

[Out]

6*ArcTan[x] - 5*Sqrt[2]*ArcTan[x/Sqrt[2]] - Log[1 + x^2]/2 + Log[2 + x^2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{2-4 x^2+x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx &=\int \left (\frac{6-x}{1+x^2}+\frac{2 (-5+x)}{2+x^2}\right ) \, dx\\ &=2 \int \frac{-5+x}{2+x^2} \, dx+\int \frac{6-x}{1+x^2} \, dx\\ &=2 \int \frac{x}{2+x^2} \, dx+6 \int \frac{1}{1+x^2} \, dx-10 \int \frac{1}{2+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=6 \tan ^{-1}(x)-5 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{2} \log \left (1+x^2\right )+\log \left (2+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0152253, size = 36, normalized size = 1. $-\frac{1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 - 4*x^2 + x^3)/((1 + x^2)*(2 + x^2)),x]

[Out]

6*ArcTan[x] - 5*Sqrt[2]*ArcTan[x/Sqrt[2]] - Log[1 + x^2]/2 + Log[2 + x^2]

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \begin{align*} 6\,\arctan \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+\ln \left ({x}^{2}+2 \right ) -5\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x)

[Out]

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

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Maxima [A]  time = 1.40449, size = 42, normalized size = 1.17 \begin{align*} -5 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 6 \, \arctan \left (x\right ) + \log \left (x^{2} + 2\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="maxima")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)

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Fricas [A]  time = 1.88212, size = 111, normalized size = 3.08 \begin{align*} -5 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 6 \, \arctan \left (x\right ) + \log \left (x^{2} + 2\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="fricas")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)

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Sympy [A]  time = 0.168758, size = 36, normalized size = 1. \begin{align*} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \log{\left (x^{2} + 2 \right )} + 6 \operatorname{atan}{\left (x \right )} - 5 \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**3-4*x**2+2)/(x**2+1)/(x**2+2),x)

[Out]

-log(x**2 + 1)/2 + log(x**2 + 2) + 6*atan(x) - 5*sqrt(2)*atan(sqrt(2)*x/2)

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Giac [A]  time = 1.055, size = 42, normalized size = 1.17 \begin{align*} -5 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 6 \, \arctan \left (x\right ) + \log \left (x^{2} + 2\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="giac")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)