3.449 \(\int \frac{1}{(-3+x) (4+x^2)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1}{26} \log \left (x^2+4\right )+\frac{1}{13} \log (3-x)-\frac{3}{26} \tan ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

(-3*ArcTan[x/2])/26 + Log[3 - x]/13 - Log[4 + x^2]/26

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Rubi [A]  time = 0.0120286, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {706, 31, 635, 203, 260} \[ -\frac{1}{26} \log \left (x^2+4\right )+\frac{1}{13} \log (3-x)-\frac{3}{26} \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((-3 + x)*(4 + x^2)),x]

[Out]

(-3*ArcTan[x/2])/26 + Log[3 - x]/13 - Log[4 + x^2]/26

Rule 706

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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}{(-3+x) \left (4+x^2\right )} \, dx &=\frac{1}{13} \int \frac{1}{-3+x} \, dx+\frac{1}{13} \int \frac{-3-x}{4+x^2} \, dx\\ &=\frac{1}{13} \log (3-x)-\frac{1}{13} \int \frac{x}{4+x^2} \, dx-\frac{3}{13} \int \frac{1}{4+x^2} \, dx\\ &=-\frac{3}{26} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{13} \log (3-x)-\frac{1}{26} \log \left (4+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0051457, size = 36, normalized size = 1.16 \[ -\frac{1}{26} \log \left ((x-3)^2+6 (x-3)+13\right )+\frac{1}{13} \log (x-3)-\frac{3}{26} \tan ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-3 + x)*(4 + x^2)),x]

[Out]

(-3*ArcTan[x/2])/26 - Log[13 + 6*(-3 + x) + (-3 + x)^2]/26 + Log[-3 + x]/13

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( -3+x \right ) }{13}}-{\frac{\ln \left ({x}^{2}+4 \right ) }{26}}-{\frac{3}{26}\arctan \left ({\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3+x)/(x^2+4),x)

[Out]

1/13*ln(-3+x)-1/26*ln(x^2+4)-3/26*arctan(1/2*x)

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Maxima [A]  time = 1.47843, size = 28, normalized size = 0.9 \begin{align*} -\frac{3}{26} \, \arctan \left (\frac{1}{2} \, x\right ) - \frac{1}{26} \, \log \left (x^{2} + 4\right ) + \frac{1}{13} \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3+x)/(x^2+4),x, algorithm="maxima")

[Out]

-3/26*arctan(1/2*x) - 1/26*log(x^2 + 4) + 1/13*log(x - 3)

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Fricas [A]  time = 1.00476, size = 80, normalized size = 2.58 \begin{align*} -\frac{3}{26} \, \arctan \left (\frac{1}{2} \, x\right ) - \frac{1}{26} \, \log \left (x^{2} + 4\right ) + \frac{1}{13} \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3+x)/(x^2+4),x, algorithm="fricas")

[Out]

-3/26*arctan(1/2*x) - 1/26*log(x^2 + 4) + 1/13*log(x - 3)

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Sympy [A]  time = 0.137788, size = 22, normalized size = 0.71 \begin{align*} \frac{\log{\left (x - 3 \right )}}{13} - \frac{\log{\left (x^{2} + 4 \right )}}{26} - \frac{3 \operatorname{atan}{\left (\frac{x}{2} \right )}}{26} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3+x)/(x**2+4),x)

[Out]

log(x - 3)/13 - log(x**2 + 4)/26 - 3*atan(x/2)/26

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Giac [A]  time = 1.11134, size = 30, normalized size = 0.97 \begin{align*} -\frac{3}{26} \, \arctan \left (\frac{1}{2} \, x\right ) - \frac{1}{26} \, \log \left (x^{2} + 4\right ) + \frac{1}{13} \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3+x)/(x^2+4),x, algorithm="giac")

[Out]

-3/26*arctan(1/2*x) - 1/26*log(x^2 + 4) + 1/13*log(abs(x - 3))