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

Optimal. Leaf size=29 \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0137334, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1814, 635, 203, 260} \[ -\frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x + x^3)/(1 + x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0114483, size = 25, normalized size = 0.86 \[ \frac{1}{2} \left (-\frac{x}{x^2+1}+\log \left (x^2+1\right )-\tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 24, normalized size = 0.8 \begin{align*} -{\frac{x}{2\,{x}^{2}+2}}-{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48918, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{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+x-1)/(x^2+1)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45813, size = 90, normalized size = 3.1 \begin{align*} -\frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.111065, size = 20, normalized size = 0.69 \begin{align*} - \frac{x}{2 x^{2} + 2} + \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11213, size = 31, normalized size = 1.07 \begin{align*} -\frac{x}{2 \,{\left (x^{2} + 1\right )}} - \frac{1}{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+x-1)/(x^2+1)^2,x, algorithm="giac")

[Out]

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