### 3.202 $$\int \frac{3+2 x}{3 x+x^3} \, dx$$

Optimal. Leaf size=28 $-\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}}$

[Out]

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

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Rubi [A]  time = 0.0224832, antiderivative size = 28, 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, 801, 635, 203, 260} $-\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 2*x)/(3*x + x^3),x]

[Out]

(2*ArcTan[x/Sqrt[3]])/Sqrt[3] + Log[x] - Log[3 + x^2]/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 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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

Mathematica [A]  time = 0.0081235, size = 28, normalized size = 1. $-\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 2*x)/(3*x + x^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 24, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+3 \right ) }{2}}+{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41322, size = 31, normalized size = 1.11 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90635, size = 85, normalized size = 3.04 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.11735, size = 29, normalized size = 1.04 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05555, size = 32, normalized size = 1.14 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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