3.437 \(\int \frac{4+x}{4 x+x^3} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.020751, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1593, 801, 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)/(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 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{4+x}{4 x+x^3} \, dx &=\int \frac{4+x}{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 = 0.0042777, 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)/(4*x + x^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, 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((4+x)/(x^3+4*x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.46112, 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((4+x)/(x^3+4*x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.01504, size = 63, normalized size = 2.74 \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((4+x)/(x^3+4*x),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.114241, 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((4+x)/(x**3+4*x),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14473, 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((4+x)/(x^3+4*x),x, algorithm="giac")

[Out]

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