Optimal. Leaf size=154 \[ \frac{1}{3 x^3}-\frac{1}{7 x^7}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.141826, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {1368, 1504, 12, 1373, 1127, 1161, 618, 204, 1164, 628} \[ \frac{1}{3 x^3}-\frac{1}{7 x^7}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1368
Rule 1504
Rule 12
Rule 1373
Rule 1127
Rule 1161
Rule 618
Rule 204
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (1+x^4+x^8\right )} \, dx &=-\frac{1}{7 x^7}+\frac{1}{7} \int \frac{-7-7 x^4}{x^4 \left (1+x^4+x^8\right )} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{21} \int -\frac{21 x^4}{1+x^4+x^8} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}+\int \frac{x^4}{1+x^4+x^8} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}+\frac{1}{2} \int \frac{x^2}{1-x^2+x^4} \, dx-\frac{1}{2} \int \frac{x^2}{1+x^2+x^4} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{4} \int \frac{1-x^2}{1-x^2+x^4} \, dx+\frac{1}{4} \int \frac{1+x^2}{1-x^2+x^4} \, dx+\frac{1}{4} \int \frac{1-x^2}{1+x^2+x^4} \, dx-\frac{1}{4} \int \frac{1+x^2}{1+x^2+x^4} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \int \frac{1+2 x}{-1-x-x^2} \, dx-\frac{1}{8} \int \frac{1-2 x}{-1+x-x^2} \, dx-\frac{1}{8} \int \frac{1}{1-x+x^2} \, dx-\frac{1}{8} \int \frac{1}{1+x+x^2} \, dx+\frac{1}{8} \int \frac{1}{1-\sqrt{3} x+x^2} \, dx+\frac{1}{8} \int \frac{1}{1+\sqrt{3} x+x^2} \, dx+\frac{\int \frac{\sqrt{3}+2 x}{-1-\sqrt{3} x-x^2} \, dx}{8 \sqrt{3}}+\frac{\int \frac{\sqrt{3}-2 x}{-1+\sqrt{3} x-x^2} \, dx}{8 \sqrt{3}}\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (1-x+x^2\right )+\frac{1}{8} \log \left (1+x+x^2\right )+\frac{\log \left (1-\sqrt{3} x+x^2\right )}{8 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} x+x^2\right )}{8 \sqrt{3}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x\right )\\ &=-\frac{1}{7 x^7}+\frac{1}{3 x^3}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}+2 x\right )-\frac{1}{8} \log \left (1-x+x^2\right )+\frac{1}{8} \log \left (1+x+x^2\right )+\frac{\log \left (1-\sqrt{3} x+x^2\right )}{8 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} x+x^2\right )}{8 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.356026, size = 171, normalized size = 1.11 \[ \frac{1}{3 x^3}-\frac{1}{7 x^7}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )}{2 \sqrt{-6+6 i \sqrt{3}}}+\frac{\left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )}{2 \sqrt{-6-6 i \sqrt{3}}}-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 119, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{7\,{x}^{7}}}+{\frac{1}{3\,{x}^{3}}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{7 \, x^{4} - 3}{21 \, x^{7}} + \frac{1}{2} \, \int \frac{x^{2}}{x^{4} - x^{2} + 1}\,{d x} + \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.62196, size = 774, normalized size = 5.03 \begin{align*} -\frac{28 \, \sqrt{6} \sqrt{3} \sqrt{2} x^{7} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2} - \sqrt{3}\right ) + 28 \, \sqrt{6} \sqrt{3} \sqrt{2} x^{7} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{-\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2} + \sqrt{3}\right ) + 7 \, \sqrt{6} \sqrt{2} x^{7} \log \left (\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2\right ) - 7 \, \sqrt{6} \sqrt{2} x^{7} \log \left (-\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2\right ) + 28 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 28 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 42 \, x^{7} \log \left (x^{2} + x + 1\right ) + 42 \, x^{7} \log \left (x^{2} - x + 1\right ) - 112 \, x^{4} + 48}{336 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.750027, size = 209, normalized size = 1.36 \begin{align*} \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} - 18432 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} - 18432 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (- 18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac{7 x^{4} - 3}{21 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{8} + x^{4} + 1\right )} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]