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

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

[Out]

-ArcTan[x/2]/2 + Log[x] + Log[4 + x^2]/2

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Rubi [A]  time = 0.0382582, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1593, 1802, 635, 203, 260} \[ \frac{1}{2} \log \left (x^2+4\right )+\log (x)-\frac{1}{2} \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(4 - x + 2*x^2)/(4*x + x^3),x]

[Out]

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

Mathematica [A]  time = 0.0046373, size = 23, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+4\right )+\log (x)-\frac{1}{2} \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[x/2]/2 + Log[x] + Log[4 + x^2]/2

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Maple [A]  time = 0.005, size = 18, normalized size = 0.8 \begin{align*} -{\frac{1}{2}\arctan \left ({\frac{x}{2}} \right ) }+\ln \left ( x \right ) +{\frac{\ln \left ({x}^{2}+4 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.40942, size = 23, normalized size = 1. \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.05247, size = 65, normalized size = 2.83 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.119764, size = 17, normalized size = 0.74 \begin{align*} \log{\left (x \right )} + \frac{\log{\left (x^{2} + 4 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + log(x**2 + 4)/2 - atan(x/2)/2

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Giac [A]  time = 1.09953, size = 24, normalized size = 1.04 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(1/2*x) + 1/2*log(x^2 + 4) + log(abs(x))