3.410 \(\int \frac{1}{x^6 \left (1+x^5+x^{10}\right )} \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{5 x^5}-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}+\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\log (x) \]

[Out]

-1/(5*x^5) - 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 = 0.103481, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{5 x^5}-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}+\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^5 + x^10)),x]

[Out]

-1/(5*x^5) - 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 = 15.3372, size = 48, normalized size = 1. \[ - \frac{\log{\left (x^{5} \right )}}{5} + \frac{\log{\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{5}}{3} + \frac{1}{3}\right ) \right )}}{15} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**10+x**5+1),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 - 1/(5*x**5)

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Mathematica [C]  time = 0.0663843, size = 208, normalized size = 4.33 \[ \frac{1}{30} \left (6 \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^7+\text{$\#$1}^5-\text{$\#$1}^4+\text{$\#$1}^3-\text{$\#$1}+1\&,\frac{4 \text{$\#$1}^7 \log (x-\text{$\#$1})-4 \text{$\#$1}^6 \log (x-\text{$\#$1})+\text{$\#$1}^5 \log (x-\text{$\#$1})+2 \text{$\#$1}^4 \log (x-\text{$\#$1})-3 \text{$\#$1}^3 \log (x-\text{$\#$1})+\text{$\#$1}^2 \log (x-\text{$\#$1})+\text{$\#$1} \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{8 \text{$\#$1}^7-7 \text{$\#$1}^6+5 \text{$\#$1}^4-4 \text{$\#$1}^3+3 \text{$\#$1}^2-1}\&\right ]-\frac{6}{x^5}+3 \log \left (x^2+x+1\right )-30 \log (x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + x^5 + x^10)),x]

[Out]

(-6/x^5 + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 30*Log[x] + 3*Log[1 + x + x^2] +
6*RootSum[1 - #1 + #1^3 - #1^4 + #1^5 - #1^7 + #1^8 & , (-Log[x - #1] + Log[x -
#1]*#1 + Log[x - #1]*#1^2 - 3*Log[x - #1]*#1^3 + 2*Log[x - #1]*#1^4 + Log[x - #
1]*#1^5 - 4*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) & ])/30

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Maple [A]  time = 0.034, size = 73, normalized size = 1.5 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{10}}-{\frac{1}{5\,{x}^{5}}}-\ln \left ( x \right ) -{\frac{\sqrt{3}}{15}\arctan \left ({\frac{2\,\sqrt{3}{x}^{5}}{3}}+{\frac{\sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 4\,{x}^{8}-4\,{x}^{7}+4\,{x}^{5}-4\,{x}^{4}+4\,{x}^{3}-4\,x+4 \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^10+x^5+1),x)

[Out]

1/10*ln(x^2+x+1)-1/5/x^5-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 [A]  time = 0.826326, size = 55, normalized size = 1.15 \[ -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - \frac{1}{5 \, x^{5}} + \frac{1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) - \frac{1}{5} \, \log \left (x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^10 + x^5 + 1)*x^6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.267185, size = 78, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (\sqrt{3} x^{5} \log \left (x^{10} + x^{5} + 1\right ) - 10 \, \sqrt{3} x^{5} \log \left (x\right ) - 2 \, x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - 2 \, \sqrt{3}\right )}}{30 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^10 + x^5 + 1)*x^6),x, algorithm="fricas")

[Out]

1/30*sqrt(3)*(sqrt(3)*x^5*log(x^10 + x^5 + 1) - 10*sqrt(3)*x^5*log(x) - 2*x^5*ar
ctan(1/3*sqrt(3)*(2*x^5 + 1)) - 2*sqrt(3))/x^5

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Sympy [A]  time = 0.609619, size = 48, normalized size = 1. \[ - \log{\left (x \right )} + \frac{\log{\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{5}}{3} + \frac{\sqrt{3}}{3} \right )}}{15} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**10+x**5+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.2824, size = 61, normalized size = 1.27 \[ -\frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) + \frac{x^{5} - 1}{5 \, x^{5}} + \frac{1}{10} \,{\rm ln}\left (x^{10} + x^{5} + 1\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^10 + x^5 + 1)*x^6),x, algorithm="giac")

[Out]

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