Optimal. Leaf size=48 \[ -\frac{1}{5 x^5}+\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}-\log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0512029, 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, Rules used = {1357, 709, 800, 634, 618, 204, 628} \[ -\frac{1}{5 x^5}+\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}-\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1357
Rule 709
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1+x^5+x^{10}\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+x+x^2\right )} \, dx,x,x^5\right )\\ &=-\frac{1}{5 x^5}+\frac{1}{5} \operatorname{Subst}\left (\int \frac{-1-x}{x \left (1+x+x^2\right )} \, dx,x,x^5\right )\\ &=-\frac{1}{5 x^5}+\frac{1}{5} \operatorname{Subst}\left (\int \left (-\frac{1}{x}+\frac{x}{1+x+x^2}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{5 x^5}-\log (x)+\frac{1}{5} \operatorname{Subst}\left (\int \frac{x}{1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac{1}{5 x^5}-\log (x)-\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^5\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac{1}{5 x^5}-\log (x)+\frac{1}{10} \log \left (1+x^5+x^{10}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^5\right )\\ &=-\frac{1}{5 x^5}-\frac{\tan ^{-1}\left (\frac{1+2 x^5}{\sqrt{3}}\right )}{5 \sqrt{3}}-\log (x)+\frac{1}{10} \log \left (1+x^5+x^{10}\right )\\ \end{align*}
Mathematica [C] time = 0.0412759, 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]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.021, size = 73, normalized size = 1.5 \begin{align*}{\frac{\ln \left ({x}^{2}+x+1 \right ) }{10}}+{\frac{\ln \left ( 4\,{x}^{8}-4\,{x}^{7}+4\,{x}^{5}-4\,{x}^{4}+4\,{x}^{3}-4\,x+4 \right ) }{10}}-{\frac{\sqrt{3}}{15}\arctan \left ({\frac{2\,\sqrt{3}{x}^{5}}{3}}+{\frac{\sqrt{3}}{3}} \right ) }-{\frac{1}{5\,{x}^{5}}}-\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.49093, size = 55, normalized size = 1.15 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59873, size = 144, normalized size = 3. \begin{align*} -\frac{2 \, \sqrt{3} x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{5} + 1\right )}\right ) - 3 \, x^{5} \log \left (x^{10} + x^{5} + 1\right ) + 30 \, x^{5} \log \left (x\right ) + 6}{30 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.219102, size = 48, normalized size = 1. \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11508, size = 61, normalized size = 1.27 \begin{align*} -\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} \, \log \left (x^{10} + x^{5} + 1\right ) - \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]