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

Optimal. Leaf size=23 \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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Rubi [A]  time = 0.0246163, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1814, 12, 203} \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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{1-3 x+2 x^2-4 x^3+x^4}{\left (1+x^2\right )^3} \, dx &=-\frac{1}{4 \left (1+x^2\right )^2}-\frac{1}{4} \int \frac{-4+16 x-4 x^2}{\left (1+x^2\right )^2} \, dx\\ &=-\frac{1}{4 \left (1+x^2\right )^2}+\frac{2}{1+x^2}+\frac{1}{8} \int \frac{8}{1+x^2} \, dx\\ &=-\frac{1}{4 \left (1+x^2\right )^2}+\frac{2}{1+x^2}+\int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{4 \left (1+x^2\right )^2}+\frac{2}{1+x^2}+\tan ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0112247, size = 23, normalized size = 1. \[ \frac{2}{x^2+1}-\frac{1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*(1 + x^2)^2) + 2/(1 + x^2) + ArcTan[x]

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Maple [A]  time = 0.004, size = 19, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( 2\,{x}^{2}+{\frac{7}{4}} \right ) }+\arctan \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^2+7/4)/(x^2+1)^2+arctan(x)

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Maxima [A]  time = 1.66992, size = 32, normalized size = 1.39 \begin{align*} \frac{8 \, x^{2} + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*x^2 + 7)/(x^4 + 2*x^2 + 1) + arctan(x)

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Fricas [A]  time = 1.46814, size = 90, normalized size = 3.91 \begin{align*} \frac{8 \, x^{2} + 4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + 7}{4 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*x^2 + 4*(x^4 + 2*x^2 + 1)*arctan(x) + 7)/(x^4 + 2*x^2 + 1)

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Sympy [A]  time = 0.122589, size = 20, normalized size = 0.87 \begin{align*} \frac{8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname{atan}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*x**2 + 7)/(4*x**4 + 8*x**2 + 4) + atan(x)

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Giac [A]  time = 1.18253, size = 26, normalized size = 1.13 \begin{align*} \frac{8 \, x^{2} + 7}{4 \,{\left (x^{2} + 1\right )}^{2}} + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*x^2 + 7)/(x^2 + 1)^2 + arctan(x)