3.1377 \(\int \frac {1}{x^8 (1+x^6)} \, dx\)

Optimal. Leaf size=90 \[ -\frac {1}{7 x^7}+\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}+\frac {1}{x}-\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{3} \tan ^{-1}(x)+\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]

[Out]

-1/7/x^7+1/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 = 0.25, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {325, 295, 634, 618, 204, 628, 203} \[ -\frac {1}{7 x^7}+\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}+\frac {1}{x}-\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{3} \tan ^{-1}(x)+\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(1 + x^6)),x]

[Out]

-1/(7*x^7) + x^(-1) - 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]) - Log[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 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]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 {1}{x^8 \left (1+x^6\right )} \, dx &=-\frac {1}{7 x^7}-\int \frac {1}{x^2 \left (1+x^6\right )} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{x}+\int \frac {x^4}{1+x^6} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{x}+\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}{1+x^2} \, dx\\ &=-\frac {1}{7 x^7}+\frac {1}{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}}\\ &=-\frac {1}{7 x^7}+\frac {1}{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 )\\ &=-\frac {1}{7 x^7}+\frac {1}{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 = 0.03, size = 84, normalized size = 0.93 \[ \frac {1}{84} \left (-\frac {12}{x^7}+7 \sqrt {3} \log \left (x^2-\sqrt {3} x+1\right )-7 \sqrt {3} \log \left (x^2+\sqrt {3} x+1\right )+\frac {84}{x}-14 \tan ^{-1}\left (\sqrt {3}-2 x\right )+28 \tan ^{-1}(x)+14 \tan ^{-1}\left (2 x+\sqrt {3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(1 + x^6)),x]

[Out]

(-12/x^7 + 84/x - 14*ArcTan[Sqrt[3] - 2*x] + 28*ArcTan[x] + 14*ArcTan[Sqrt[3] + 2*x] + 7*Sqrt[3]*Log[1 - Sqrt[
3]*x + x^2] - 7*Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/84

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fricas [A]  time = 0.91, size = 115, normalized size = 1.28 \[ -\frac {7 \, \sqrt {3} x^{7} \log \left (x^{2} + \sqrt {3} x + 1\right ) - 7 \, \sqrt {3} x^{7} \log \left (x^{2} - \sqrt {3} x + 1\right ) - 28 \, x^{7} \arctan \relax (x) + 28 \, x^{7} \arctan \left (-2 \, x + \sqrt {3} + 2 \, \sqrt {x^{2} - \sqrt {3} x + 1}\right ) + 28 \, x^{7} \arctan \left (-2 \, x - \sqrt {3} + 2 \, \sqrt {x^{2} + \sqrt {3} x + 1}\right ) - 84 \, x^{6} + 12}{84 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(x^6+1),x, algorithm="fricas")

[Out]

-1/84*(7*sqrt(3)*x^7*log(x^2 + sqrt(3)*x + 1) - 7*sqrt(3)*x^7*log(x^2 - sqrt(3)*x + 1) - 28*x^7*arctan(x) + 28
*x^7*arctan(-2*x + sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1)) + 28*x^7*arctan(-2*x - sqrt(3) + 2*sqrt(x^2 + sqrt(3
)*x + 1)) - 84*x^6 + 12)/x^7

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giac [A]  time = 0.16, size = 39, normalized size = 0.43 \[ \frac {7 \, x^{6} - 1}{7 \, x^{7}} + \frac {1}{3} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(x^6+1),x, algorithm="giac")

[Out]

1/7*(7*x^6 - 1)/x^7 + 1/3*arctan(2*x + sqrt(3)) + 1/3*arctan(2*x - sqrt(3)) + 1/3*arctan(x)

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maple [A]  time = 0.02, size = 69, normalized size = 0.77 \[ \frac {\arctan \relax (x )}{3}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{6}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{6}+\frac {\sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, x +1\right )}{12}-\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, x +1\right )}{12}+\frac {1}{x}-\frac {1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(x^6+1),x)

[Out]

-1/7/x^7+1/x+1/3*arctan(x)+1/6*arctan(2*x-3^(1/2))+1/6*arctan(2*x+3^(1/2))+1/12*3^(1/2)*ln(x^2-3^(1/2)*x+1)-1/
12*3^(1/2)*ln(x^2+3^(1/2)*x+1)

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maxima [A]  time = 2.39, size = 72, normalized size = 0.80 \[ -\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {7 \, x^{6} - 1}{7 \, x^{7}} + \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(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) + 1/7*(7*x^6 - 1)/x^7 + 1/6*arc
tan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1/3*arctan(x)

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mupad [B]  time = 1.05, size = 62, normalized size = 0.69 \[ \frac {\mathrm {atan}\relax (x)}{3}-\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {x^6-\frac {1}{7}}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(x^6 + 1)),x)

[Out]

atan(x)/3 - atan((2*x)/(3^(1/2)*1i - 1))*((3^(1/2)*1i)/6 + 1/6) - atan((2*x)/(3^(1/2)*1i + 1))*((3^(1/2)*1i)/6
- 1/6) + (x^6 - 1/7)/x^7

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sympy [A]  time = 0.46, size = 80, normalized size = 0.89 \[ \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\relax (x )}}{3} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{6} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{6} + \frac {7 x^{6} - 1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(x**6+1),x)

[Out]

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 + (7*x**6 - 1)/(7*x**7)

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