Optimal. Leaf size=102 \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{x}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0556487, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846, Rules used = {325, 301, 297, 1162, 617, 204, 1165, 628, 298, 203, 206} \[ -\frac{\log \left (x^2-\sqrt{2} x+1\right )}{8 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{8 \sqrt{2}}-\frac{1}{x}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 301
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (1-x^8\right )} \, dx &=-\frac{1}{x}+\int \frac{x^6}{1-x^8} \, dx\\ &=-\frac{1}{x}+\frac{1}{2} \int \frac{x^2}{1-x^4} \, dx-\frac{1}{2} \int \frac{x^2}{1+x^4} \, dx\\ &=-\frac{1}{x}+\frac{1}{4} \int \frac{1}{1-x^2} \, dx-\frac{1}{4} \int \frac{1}{1+x^2} \, dx+\frac{1}{4} \int \frac{1-x^2}{1+x^4} \, dx-\frac{1}{4} \int \frac{1+x^2}{1+x^4} \, dx\\ &=-\frac{1}{x}-\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x)-\frac{1}{8} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx-\frac{1}{8} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx-\frac{\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{8 \sqrt{2}}-\frac{\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{8 \sqrt{2}}\\ &=-\frac{1}{x}-\frac{1}{4} \tan ^{-1}(x)+\frac{1}{4} \tanh ^{-1}(x)-\frac{\log \left (1-\sqrt{2} x+x^2\right )}{8 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} x+x^2\right )}{8 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{4 \sqrt{2}}\\ &=-\frac{1}{x}-\frac{1}{4} \tan ^{-1}(x)+\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} x\right )}{4 \sqrt{2}}+\frac{1}{4} \tanh ^{-1}(x)-\frac{\log \left (1-\sqrt{2} x+x^2\right )}{8 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} x+x^2\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0325884, size = 109, normalized size = 1.07 \[ -\frac{\sqrt{2} x \log \left (x^2-\sqrt{2} x+1\right )-\sqrt{2} x \log \left (x^2+\sqrt{2} x+1\right )+2 x \log (1-x)-2 x \log (x+1)+4 x \tan ^{-1}(x)-2 \sqrt{2} x \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \sqrt{2} x \tan ^{-1}\left (\sqrt{2} x+1\right )+16}{16 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 79, normalized size = 0.8 \begin{align*} -{\frac{\arctan \left ( x \right ) }{4}}+{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\sqrt{2}}{16}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) }-{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{8}}-{x}^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46255, size = 126, normalized size = 1.24 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{x} - \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.38342, size = 382, normalized size = 3.75 \begin{align*} \frac{4 \, \sqrt{2} x \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) + 4 \, \sqrt{2} x \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) + \sqrt{2} x \log \left (x^{2} + \sqrt{2} x + 1\right ) - \sqrt{2} x \log \left (x^{2} - \sqrt{2} x + 1\right ) - 4 \, x \arctan \left (x\right ) + 2 \, x \log \left (x + 1\right ) - 2 \, x \log \left (x - 1\right ) - 16}{16 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 106.584, size = 49, normalized size = 0.48 \begin{align*} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} + \frac{i \log{\left (x - i \right )}}{8} - \frac{i \log{\left (x + i \right )}}{8} - \operatorname{RootSum}{\left (4096 t^{4} + 1, \left ( t \mapsto t \log{\left (- 2097152 t^{7} + x \right )} \right )\right )} - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15946, size = 128, normalized size = 1.25 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{16} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{1}{x} - \frac{1}{4} \, \arctan \left (x\right ) + \frac{1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]