3.309 \(\int \frac{1-3 x+2 x^2-x^3}{(1+x^2)^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{2-x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\frac{3}{2} \tan ^{-1}(x) \]

[Out]

(2 - x)/(2*(1 + x^2)) + (3*ArcTan[x])/2 - Log[1 + x^2]/2

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Rubi [A]  time = 0.0167461, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1814, 635, 203, 260} \[ \frac{2-x}{2 \left (x^2+1\right )}-\frac{1}{2} \log \left (x^2+1\right )+\frac{3}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2 - x)/(2*(1 + x^2)) + (3*ArcTan[x])/2 - Log[1 + x^2]/2

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

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

Mathematica [A]  time = 0.0111377, size = 30, normalized size = 0.91 \[ \frac{1}{2} \left (\frac{2-x}{x^2+1}-\log \left (x^2+1\right )+3 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2 - x)/(1 + x^2) + 3*ArcTan[x] - Log[1 + x^2])/2

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{{x}^{2}+1} \left ({\frac{x}{2}}-1 \right ) }-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{3\,\arctan \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.62154, size = 34, normalized size = 1.03 \begin{align*} -\frac{x - 2}{2 \,{\left (x^{2} + 1\right )}} + \frac{3}{2} \, \arctan \left (x\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+2*x^2-3*x+1)/(x^2+1)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56371, size = 97, normalized size = 2.94 \begin{align*} \frac{3 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - x + 2}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.116215, size = 24, normalized size = 0.73 \begin{align*} - \frac{x - 2}{2 x^{2} + 2} - \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{3 \operatorname{atan}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**3+2*x**2-3*x+1)/(x**2+1)**2,x)

[Out]

-(x - 2)/(2*x**2 + 2) - log(x**2 + 1)/2 + 3*atan(x)/2

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Giac [A]  time = 1.38542, size = 34, normalized size = 1.03 \begin{align*} -\frac{x - 2}{2 \,{\left (x^{2} + 1\right )}} + \frac{3}{2} \, \arctan \left (x\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+2*x^2-3*x+1)/(x^2+1)^2,x, algorithm="giac")

[Out]

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