3.1363 \(\int \frac {x^6}{1+x^6} \, dx\)

Optimal. Leaf size=81 \[ \frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}+x+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]

[Out]

x-1/3*arctan(x)-1/6*arctan(2*x-3^(1/2))-1/6*arctan(2*x+3^(1/2))+1/12*ln(1+x^2-x*3^(1/2))*3^(1/2)-1/12*ln(1+x^2
+x*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {321, 209, 634, 618, 204, 628, 203} \[ \frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}+x+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6/(1 + x^6),x]

[Out]

x + ArcTan[Sqrt[3] - 2*x]/6 - ArcTan[x]/3 - ArcTan[Sqrt[3] + 2*x]/6 + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - L
og[1 + Sqrt[3]*x + x^2]/(4*Sqrt[3])

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 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 76, normalized size = 0.94 \[ \frac {1}{12} \left (\sqrt {3} \log \left (x^2-\sqrt {3} x+1\right )-\sqrt {3} \log \left (x^2+\sqrt {3} x+1\right )+12 x+2 \tan ^{-1}\left (\sqrt {3}-2 x\right )-4 \tan ^{-1}(x)-2 \tan ^{-1}\left (2 x+\sqrt {3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(1 + x^6),x]

[Out]

(12*x + 2*ArcTan[Sqrt[3] - 2*x] - 4*ArcTan[x] - 2*ArcTan[Sqrt[3] + 2*x] + Sqrt[3]*Log[1 - Sqrt[3]*x + x^2] - S
qrt[3]*Log[1 + Sqrt[3]*x + x^2])/12

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fricas [A]  time = 0.89, size = 90, normalized size = 1.11 \[ -\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + x - \frac {1}{3} \, \arctan \relax (x) + \frac {1}{3} \, \arctan \left (-2 \, x + \sqrt {3} + 2 \, \sqrt {x^{2} - \sqrt {3} x + 1}\right ) + \frac {1}{3} \, \arctan \left (-2 \, x - \sqrt {3} + 2 \, \sqrt {x^{2} + \sqrt {3} x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + x - 1/3*arctan(x) + 1/3*arcta
n(-2*x + sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1)) + 1/3*arctan(-2*x - sqrt(3) + 2*sqrt(x^2 + sqrt(3)*x + 1))

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giac [A]  time = 0.16, size = 61, normalized size = 0.75 \[ -\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + x - \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{3} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + x - 1/6*arctan(2*x + sqrt(3))
- 1/6*arctan(2*x - sqrt(3)) - 1/3*arctan(x)

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maple [A]  time = 0.04, size = 62, normalized size = 0.77 \[ x -\frac {\arctan \relax (x )}{3}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{6}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{6}+\frac {\sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, x +1\right )}{12}-\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, x +1\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-1/3*arctan(x)-1/6*arctan(2*x-3^(1/2))-1/6*arctan(2*x+3^(1/2))+1/12*ln(1+x^2-3^(1/2)*x)*3^(1/2)-1/12*ln(1+x^2
+3^(1/2)*x)*3^(1/2)

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maxima [A]  time = 2.38, size = 61, normalized size = 0.75 \[ -\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + x - \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{3} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + x - 1/6*arctan(2*x + sqrt(3))
- 1/6*arctan(2*x - sqrt(3)) - 1/3*arctan(x)

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mupad [B]  time = 0.11, size = 88, normalized size = 1.09 \[ x-\frac {\mathrm {atan}\relax (x)}{3}-\mathrm {atan}\left (\frac {x}{-1+\sqrt {3}\,1{}\mathrm {i}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\mathrm {atan}\left (\frac {x}{1+\sqrt {3}\,1{}\mathrm {i}}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^6 + 1),x)

[Out]

x - atan(x)/3 - atan(x/(3^(1/2)*1i - 1) + (3^(1/2)*x*1i)/(3^(1/2)*1i - 1))*((3^(1/2)*1i)/6 + 1/6) - atan(x/(3^
(1/2)*1i + 1) - (3^(1/2)*x*1i)/(3^(1/2)*1i + 1))*((3^(1/2)*1i)/6 - 1/6)

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sympy [A]  time = 0.42, size = 70, normalized size = 0.86 \[ x + \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} - \frac {\operatorname {atan}{\relax (x )}}{3} - \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{6} - \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 - atan(x)/3 - atan(2*x - sqrt(
3))/6 - atan(2*x + sqrt(3))/6

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