Optimal. Leaf size=166 \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.142119, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {202, 634, 618, 204, 628, 31} \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{1-x^7} \, dx &=\frac{2}{7} \int \frac{1+x \cos \left (\frac{\pi }{7}\right )}{1+x^2+2 x \cos \left (\frac{\pi }{7}\right )} \, dx+\frac{2}{7} \int \frac{1+x \sin \left (\frac{\pi }{14}\right )}{1+x^2+2 x \sin \left (\frac{\pi }{14}\right )} \, dx+\frac{2}{7} \int \frac{1-x \sin \left (\frac{3 \pi }{14}\right )}{1+x^2-2 x \sin \left (\frac{3 \pi }{14}\right )} \, dx+\frac{1}{7} \int \frac{1}{1-x} \, dx\\ &=-\frac{1}{7} \log (1-x)+\frac{1}{7} \left (2 \cos ^2\left (\frac{\pi }{14}\right )\right ) \int \frac{1}{1+x^2+2 x \sin \left (\frac{\pi }{14}\right )} \, dx+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \int \frac{2 x+2 \cos \left (\frac{\pi }{7}\right )}{1+x^2+2 x \cos \left (\frac{\pi }{7}\right )} \, dx+\frac{1}{7} \left (2 \cos ^2\left (\frac{3 \pi }{14}\right )\right ) \int \frac{1}{1+x^2-2 x \sin \left (\frac{3 \pi }{14}\right )} \, dx+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \int \frac{2 x+2 \sin \left (\frac{\pi }{14}\right )}{1+x^2+2 x \sin \left (\frac{\pi }{14}\right )} \, dx+\frac{1}{7} \left (2 \sin ^2\left (\frac{\pi }{7}\right )\right ) \int \frac{1}{1+x^2+2 x \cos \left (\frac{\pi }{7}\right )} \, dx-\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \int \frac{2 x-2 \sin \left (\frac{3 \pi }{14}\right )}{1+x^2-2 x \sin \left (\frac{3 \pi }{14}\right )} \, dx\\ &=-\frac{1}{7} \log (1-x)+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac{\pi }{7}\right )\right )+\frac{1}{7} \log \left (1+x^2+2 x \sin \left (\frac{\pi }{14}\right )\right ) \sin \left (\frac{\pi }{14}\right )-\frac{1}{7} \log \left (1+x^2-2 x \sin \left (\frac{3 \pi }{14}\right )\right ) \sin \left (\frac{3 \pi }{14}\right )-\frac{1}{7} \left (4 \cos ^2\left (\frac{\pi }{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \cos ^2\left (\frac{\pi }{14}\right )} \, dx,x,2 x+2 \sin \left (\frac{\pi }{14}\right )\right )-\frac{1}{7} \left (4 \cos ^2\left (\frac{3 \pi }{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \cos ^2\left (\frac{3 \pi }{14}\right )} \, dx,x,2 x-2 \sin \left (\frac{3 \pi }{14}\right )\right )-\frac{1}{7} \left (4 \sin ^2\left (\frac{\pi }{7}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \sin ^2\left (\frac{\pi }{7}\right )} \, dx,x,2 x+2 \cos \left (\frac{\pi }{7}\right )\right )\\ &=\frac{2}{7} \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \cos \left (\frac{\pi }{14}\right )+\frac{2}{7} \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right ) \cos \left (\frac{3 \pi }{14}\right )-\frac{1}{7} \log (1-x)+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (1+x^2+2 x \cos \left (\frac{\pi }{7}\right )\right )+\frac{1}{7} \log \left (1+x^2+2 x \sin \left (\frac{\pi }{14}\right )\right ) \sin \left (\frac{\pi }{14}\right )+\frac{2}{7} \tan ^{-1}\left (\left (x+\cos \left (\frac{\pi }{7}\right )\right ) \csc \left (\frac{\pi }{7}\right )\right ) \sin \left (\frac{\pi }{7}\right )-\frac{1}{7} \log \left (1+x^2-2 x \sin \left (\frac{3 \pi }{14}\right )\right ) \sin \left (\frac{3 \pi }{14}\right )\\ \end{align*}
Mathematica [A] time = 0.0042524, size = 166, normalized size = 1. \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.007, size = 89, normalized size = 0.5 \begin{align*}{\frac{1}{7}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{5}+{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+{{\it \_Z}}^{2}+{\it \_Z}+1 \right ) }{\frac{ \left ({{\it \_R}}^{5}+2\,{{\it \_R}}^{4}+3\,{{\it \_R}}^{3}+4\,{{\it \_R}}^{2}+5\,{\it \_R}+6 \right ) \ln \left ( x-{\it \_R} \right ) }{6\,{{\it \_R}}^{5}+5\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+2\,{\it \_R}+1}}}-{\frac{\ln \left ( -1+x \right ) }{7}} \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}{7} \, \int \frac{x^{5} + 2 \, x^{4} + 3 \, x^{3} + 4 \, x^{2} + 5 \, x + 6}{x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1}\,{d x} - \frac{1}{7} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 25.8919, size = 1018, normalized size = 6.13 \begin{align*} \frac{1}{14} \,{\left (\sqrt{-0.7530203962825330? + 0.?e-36 \sqrt{-1}} + 1.801937735804839? + 0.?e-36 \sqrt{-1}\right )} \log \left (2 \, x + \sqrt{-0.7530203962825330? + 0.?e-36 \sqrt{-1}} + 1.801937735804839? + 0.?e-36 \sqrt{-1}\right ) - \frac{1}{14} \,{\left (\sqrt{-0.7530203962825330? + 0.?e-36 \sqrt{-1}} - 1.801937735804839? + 0.?e-36 \sqrt{-1}\right )} \log \left (2 \, x - \sqrt{-0.7530203962825330? + 0.?e-36 \sqrt{-1}} + 1.801937735804839? + 0.?e-36 \sqrt{-1}\right ) + \left (0.03178870485090206? + 0.1392754160259748? \sqrt{-1}\right ) \, \log \left (x + 0.2225209339563144? + 0.9749279121818236? \sqrt{-1}\right ) + \left (0.03178870485090206? - 0.1392754160259748? \sqrt{-1}\right ) \, \log \left (x + 0.2225209339563144? - 0.9749279121818236? \sqrt{-1}\right ) - \left (0.08906997169410479? - 0.11169021178114711? \sqrt{-1}\right ) \, \log \left (x - 0.6234898018587335? + 0.7818314824680299? \sqrt{-1}\right ) - \left (0.08906997169410479? + 0.11169021178114711? \sqrt{-1}\right ) \, \log \left (x - 0.6234898018587335? - 0.7818314824680299? \sqrt{-1}\right ) - \frac{1}{7} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.179433, size = 46, normalized size = 0.28 \begin{align*} - \frac{\log{\left (x - 1 \right )}}{7} - \operatorname{RootSum}{\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log{\left (- 7 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.55063, size = 171, normalized size = 1.03 \begin{align*} \frac{1}{7} \, \cos \left (\frac{3}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac{3}{7} \, \pi \right ) + 1\right ) - \frac{1}{7} \, \cos \left (\frac{2}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac{2}{7} \, \pi \right ) + 1\right ) + \frac{1}{7} \, \cos \left (\frac{1}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac{1}{7} \, \pi \right ) + 1\right ) + \frac{2}{7} \, \arctan \left (\frac{x + \cos \left (\frac{3}{7} \, \pi \right )}{\sin \left (\frac{3}{7} \, \pi \right )}\right ) \sin \left (\frac{3}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x - \cos \left (\frac{2}{7} \, \pi \right )}{\sin \left (\frac{2}{7} \, \pi \right )}\right ) \sin \left (\frac{2}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x + \cos \left (\frac{1}{7} \, \pi \right )}{\sin \left (\frac{1}{7} \, \pi \right )}\right ) \sin \left (\frac{1}{7} \, \pi \right ) - \frac{1}{7} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]