3.332 \(\int \frac {1-3 x+2 x^2-4 x^3+x^4}{1+3 x^2+3 x^4+x^6} \, dx\)

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

[Out]

-1/4/(x^2+1)^2+2/(x^2+1)+arctan(x)

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Rubi [A]  time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2073, 261, 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 + 3*x^2 + 3*x^4 + x^6),x]

[Out]

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

Rule 261

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

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x + 2*x^2 - 4*x^3 + x^4)/(1 + 3*x^2 + 3*x^4 + x^6),x]

[Out]

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

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fricas [A]  time = 1.24, size = 35, normalized size = 1.52 \[ \frac {8 \, x^{2} + 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x) + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4*x^3+2*x^2-3*x+1)/(x^6+3*x^4+3*x^2+1),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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giac [A]  time = 0.27, size = 19, normalized size = 0.83 \[ \frac {8 \, x^{2} + 7}{4 \, {\left (x^{2} + 1\right )}^{2}} + \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4*x^3+2*x^2-3*x+1)/(x^6+3*x^4+3*x^2+1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.83 \[ \arctan \relax (x )+\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-4*x^3+2*x^2-3*x+1)/(x^6+3*x^4+3*x^2+1),x)

[Out]

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

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maxima [A]  time = 1.42, size = 24, normalized size = 1.04 \[ \frac {8 \, x^{2} + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-4*x^3+2*x^2-3*x+1)/(x^6+3*x^4+3*x^2+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 23, normalized size = 1.00 \[ \mathrm {atan}\relax (x)+\frac {2\,x^2+\frac {7}{4}}{x^4+2\,x^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - 3*x - 4*x^3 + x^4 + 1)/(3*x^2 + 3*x^4 + x^6 + 1),x)

[Out]

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

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sympy [A]  time = 0.13, size = 20, normalized size = 0.87 \[ \frac {8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname {atan}{\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-4*x**3+2*x**2-3*x+1)/(x**6+3*x**4+3*x**2+1),x)

[Out]

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

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