3.146 \(\int \frac{1+x^4}{1+x^6} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{2}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.425398, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {1876, 209, 634, 618, 204, 628, 203, 295} \[ -\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{2}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 + x^6),x]

[Out]

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

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1+x^4}{1+x^6} \, dx &=\int \left (\frac{1}{1+x^6}+\frac{x^4}{1+x^6}\right ) \, dx\\ &=\int \frac{1}{1+x^6} \, dx+\int \frac{x^4}{1+x^6} \, dx\\ &=\frac{1}{3} \int \frac{1-\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx+\frac{1}{3} \int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx+\frac{1}{3} \int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx+\frac{1}{3} \int \frac{1+\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx+\frac{2}{3} \int \frac{1}{1+x^2} \, dx\\ &=\frac{2}{3} \tan ^{-1}(x)+2 \left (\frac{1}{12} \int \frac{1}{1-\sqrt{3} x+x^2} \, dx\right )+2 \left (\frac{1}{12} \int \frac{1}{1+\sqrt{3} x+x^2} \, dx\right )\\ &=\frac{2}{3} \tan ^{-1}(x)-2 \left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x\right )\right )-2 \left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x\right )\right )\\ &=-\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{2}{3} \tan ^{-1}(x)+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}+2 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0069387, size = 21, normalized size = 0.6 \[ \frac{2}{3} \tan ^{-1}(x)-\frac{1}{3} \tan ^{-1}\left (\frac{x}{x^2-1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(1 + x^6),x]

[Out]

(2*ArcTan[x])/3 - ArcTan[x/(-1 + x^2)]/3

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Maple [A]  time = 0.024, size = 28, normalized size = 0.8 \begin{align*}{\frac{2\,\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{3}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.41201, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{3} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{2}{3} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^6+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.99997, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{3} \, \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^6+1),x, algorithm="fricas")

[Out]

1/3*arctan(x^3) + arctan(x)

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Sympy [A]  time = 0.110474, size = 8, normalized size = 0.23 \begin{align*} \operatorname{atan}{\left (x \right )} + \frac{\operatorname{atan}{\left (x^{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**6+1),x)

[Out]

atan(x) + atan(x**3)/3

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Giac [A]  time = 1.0464, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{3} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{2}{3} \, \arctan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^6+1),x, algorithm="giac")

[Out]

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