Optimal. Leaf size=919 \[ \text{result too large to display} \]
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Rubi [A] time = 0.899297, antiderivative size = 919, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 20, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.429, Rules used = {6171, 105, 63, 331, 299, 1122, 1169, 634, 618, 204, 628, 93, 214, 212, 206, 203, 211, 1165, 1162, 617} \[ -\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2+\sqrt{2}}}\right )-\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2-\sqrt{2}}}\right )+\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+1\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+1\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+1\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}+1\right )-\frac{\log \left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 105
Rule 63
Rule 331
Rule 299
Rule 1122
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 1162
Rule 617
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x \sqrt [8]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/8}} \, dx,x,\frac{1}{x}\right )\\ &=8 \operatorname{Subst}\left (\int \frac{x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-\frac{1}{a x}}\right )-8 \operatorname{Subst}\left (\int \frac{1}{-1+x^8} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+4 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+8 \operatorname{Subst}\left (\int \frac{x^6}{1+x^8} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )\\ &=2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}-\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{2} x^2}{1-\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{2} x^2}{1+\sqrt{2} x^2+x^4} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )\\ &=2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )-\sqrt{2-\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}-\left (1+\sqrt{2}\right ) x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\sqrt{2-\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}+\left (1+\sqrt{2}\right ) x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\sqrt{2+\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}-\left (1-\sqrt{2}\right ) x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\sqrt{2+\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}+\left (1-\sqrt{2}\right ) x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )\\ &=-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{1}{2} \sqrt{2-\sqrt{2}} \operatorname{Subst}\left (\int \frac{-\sqrt{2-\sqrt{2}}+2 x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2-\sqrt{2}}+2 x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \left (-2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \left (-2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \operatorname{Subst}\left (\int \frac{-\sqrt{2+\sqrt{2}}+2 x}{1-\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \operatorname{Subst}\left (\int \frac{\sqrt{2+\sqrt{2}}+2 x}{1+\sqrt{2+\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\frac{1}{2} \left (2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\frac{1}{2} \left (2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx,x,\frac{\sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )\\ &=-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}-\left (2-\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,-\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\left (2-\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{2}-x^2} \, dx,x,\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\left (2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,-\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\left (2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )\\ &=-\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2+\sqrt{2}}}\right )-\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2-\sqrt{2}}}\right )+\sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2+\sqrt{2}}}\right )+\sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}}{\sqrt{2-\sqrt{2}}}\right )-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left (1+\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{1+\frac{1}{a x}}}\right )-\frac{\log \left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0390659, size = 30, normalized size = 0.03 \[ \frac{16}{9} e^{\frac{9}{4} \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{9}{16},1,\frac{25}{16},e^{4 \coth ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [8]{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13455, size = 6747, normalized size = 7.34 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25315, size = 878, normalized size = 0.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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