### 3.170 $$\int \frac{1+x^2+x^4}{(1+x^2) (4+x^2)^2} \, dx$$

Optimal. Leaf size=29 $-\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x)$

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

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Rubi [A]  time = 0.113411, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {6725, 203, 199} $-\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x^2 + x^4)/((1 + x^2)*(4 + x^2)^2),x]

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rubi steps

\begin{align*} \int \frac{1+x^2+x^4}{\left (1+x^2\right ) \left (4+x^2\right )^2} \, dx &=\int \left (\frac{1}{9 \left (1+x^2\right )}-\frac{13}{3 \left (4+x^2\right )^2}+\frac{8}{9 \left (4+x^2\right )}\right ) \, dx\\ &=\frac{1}{9} \int \frac{1}{1+x^2} \, dx+\frac{8}{9} \int \frac{1}{4+x^2} \, dx-\frac{13}{3} \int \frac{1}{\left (4+x^2\right )^2} \, dx\\ &=-\frac{13 x}{24 \left (4+x^2\right )}+\frac{4}{9} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x)-\frac{13}{24} \int \frac{1}{4+x^2} \, dx\\ &=-\frac{13 x}{24 \left (4+x^2\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0161005, size = 29, normalized size = 1. $-\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + x^2 + x^4)/((1 + x^2)*(4 + x^2)^2),x]

[Out]

(-13*x)/(24*(4 + x^2)) + (25*ArcTan[x/2])/144 + ArcTan[x]/9

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Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{24\,{x}^{2}+96}}+{\frac{25}{144}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/24*x/(x^2+4)+25/144*arctan(1/2*x)+1/9*arctan(x)

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Maxima [A]  time = 1.40065, size = 28, normalized size = 0.97 \begin{align*} -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/24*x/(x^2 + 4) + 25/144*arctan(1/2*x) + 1/9*arctan(x)

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Fricas [A]  time = 1.90528, size = 105, normalized size = 3.62 \begin{align*} \frac{25 \,{\left (x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, x\right ) + 16 \,{\left (x^{2} + 4\right )} \arctan \left (x\right ) - 78 \, x}{144 \,{\left (x^{2} + 4\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(25*(x^2 + 4)*arctan(1/2*x) + 16*(x^2 + 4)*arctan(x) - 78*x)/(x^2 + 4)

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Sympy [A]  time = 0.156617, size = 22, normalized size = 0.76 \begin{align*} - \frac{13 x}{24 x^{2} + 96} + \frac{25 \operatorname{atan}{\left (\frac{x}{2} \right )}}{144} + \frac{\operatorname{atan}{\left (x \right )}}{9} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13*x/(24*x**2 + 96) + 25*atan(x/2)/144 + atan(x)/9

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Giac [A]  time = 1.04939, size = 28, normalized size = 0.97 \begin{align*} -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/24*x/(x^2 + 4) + 25/144*arctan(1/2*x) + 1/9*arctan(x)