3.323 \(\int \frac{2 x+x^4}{1+x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

[Out]

-x + x^3/3 + ArcTan[x] + Log[1 + x^2]

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Rubi [A]  time = 0.0267939, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1593, 1802, 635, 203, 260} \[ \frac{x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(2*x + x^4)/(1 + x^2),x]

[Out]

-x + x^3/3 + ArcTan[x] + Log[1 + x^2]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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

Mathematica [A]  time = 0.0049293, size = 19, normalized size = 1. \[ \frac{x^3}{3}+\log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + x^4)/(1 + x^2),x]

[Out]

-x + x^3/3 + ArcTan[x] + Log[1 + x^2]

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Maple [A]  time = 0.001, size = 18, normalized size = 1. \begin{align*} -x+{\frac{{x}^{3}}{3}}+\arctan \left ( x \right ) +\ln \left ({x}^{2}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46024, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{3} \, x^{3} - x + \arctan \left (x\right ) + \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.46225, size = 54, normalized size = 2.84 \begin{align*} \frac{1}{3} \, x^{3} - x + \arctan \left (x\right ) + \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.090904, size = 15, normalized size = 0.79 \begin{align*} \frac{x^{3}}{3} - x + \log{\left (x^{2} + 1 \right )} + \operatorname{atan}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 - x + log(x**2 + 1) + atan(x)

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Giac [A]  time = 1.2895, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{3} \, x^{3} - x + \arctan \left (x\right ) + \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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