3.411 \(\int \frac{1}{x+x^6+x^{11}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}-\frac{1}{10} \log \left (x^{10}+x^5+1\right )+\log (x) \]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x^5 + x^10]/10

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Rubi [A]  time = 0.066934, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ -\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}-\frac{1}{10} \log \left (x^{10}+x^5+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(x + x^6 + x^11)^(-1),x]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x^5 + x^10]/10

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Rubi in Sympy [A]  time = 13.4016, size = 41, normalized size = 1.05 \[ \frac{\log{\left (x^{5} \right )}}{5} - \frac{\log{\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{5}}{3} + \frac{1}{3}\right ) \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**11+x**6+x),x)

[Out]

log(x**5)/5 - log(x**10 + x**5 + 1)/10 - sqrt(3)*atan(sqrt(3)*(2*x**5/3 + 1/3))/
15

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Mathematica [C]  time = 0.0274597, size = 197, normalized size = 5.05 \[ -\frac{1}{5} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\&,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-3 \text{$\#$1}^6 \log (x-\text{$\#$1})-\text{$\#$1}^5 \log (x-\text{$\#$1})+3 \text{$\#$1}^4 \log (x-\text{$\#$1})-\text{$\#$1}^3 \log (x-\text{$\#$1})+2 \text{$\#$1}^2 \log (x-\text{$\#$1})-\text{$\#$1} \log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\&\right ]-\frac{1}{10} \log \left (x^2+x+1\right )+\log (x)+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{5 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x + x^6 + x^11)^(-1),x]

[Out]

ArcTan[(1 + 2*x)/Sqrt[3]]/(5*Sqrt[3]) + Log[x] - Log[1 + x + x^2]/10 - RootSum[1
- #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-(Log[x - #1]*#1) + 2*Log[x - #1]*
#1^2 - Log[x - #1]*#1^3 + 3*Log[x - #1]*#1^4 - Log[x - #1]*#1^5 - 3*Log[x - #1]*
#1^6 + 4*Log[x - #1]*#1^7)/(-1 + 3*#1^2 - 4*#1^3 + 5*#1^4 - 7*#1^6 + 8*#1^7) & ]
/5

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Maple [B]  time = 0.017, size = 66, normalized size = 1.7 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{10}}+\ln \left ( x \right ) -{\frac{\sqrt{3}}{15}\arctan \left ({\frac{2\,\sqrt{3}{x}^{5}}{3}}+{\frac{\sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 4\,{x}^{8}-4\,{x}^{7}+4\,{x}^{5}-4\,{x}^{4}+4\,{x}^{3}-4\,x+4 \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^11+x^6+x),x)

[Out]

-1/10*ln(x^2+x+1)+ln(x)-1/15*3^(1/2)*arctan(2/3*3^(1/2)*x^5+1/3*3^(1/2))-1/10*ln
(4*x^8-4*x^7+4*x^5-4*x^4+4*x^3-4*x+4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{5} \, \int \frac{4 \, x^{7} - 3 \, x^{6} - x^{5} + 3 \, x^{4} - x^{3} + 2 \, x^{2} - x}{x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1}\,{d x} - \frac{1}{10} \, \log \left (x^{2} + x + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^11 + x^6 + x),x, algorithm="maxima")

[Out]

1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/5*integrate((4*x^7 - 3*x^6 - x^5
+ 3*x^4 - x^3 + 2*x^2 - x)/(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1), x) - 1/10*log(
x^2 + x + 1) + log(x)

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Fricas [A]  time = 0.255054, size = 55, normalized size = 1.41 \[ -\frac{1}{30} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{10} + x^{5} + 1\right ) - 10 \, \sqrt{3} \log \left (x\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^11 + x^6 + x),x, algorithm="fricas")

[Out]

-1/30*sqrt(3)*(sqrt(3)*log(x^10 + x^5 + 1) - 10*sqrt(3)*log(x) + 2*arctan(1/3*sq
rt(3)*(2*x^5 + 1)))

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Sympy [A]  time = 0.427512, size = 41, normalized size = 1.05 \[ \log{\left (x \right )} - \frac{\log{\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{5}}{3} + \frac{\sqrt{3}}{3} \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**11+x**6+x),x)

[Out]

log(x) - log(x**10 + x**5 + 1)/10 - sqrt(3)*atan(2*sqrt(3)*x**5/3 + sqrt(3)/3)/1
5

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GIAC/XCAS [A]  time = 0.266234, size = 45, normalized size = 1.15 \[ -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - \frac{1}{10} \,{\rm ln}\left (x^{10} + x^{5} + 1\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^11 + x^6 + x),x, algorithm="giac")

[Out]

-1/15*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^5 + 1)) - 1/10*ln(x^10 + x^5 + 1) + ln(abs
(x))