Optimal. Leaf size=676 \[ a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]
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Rubi [A] time = 0.605244, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {6171, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )+\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left (\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+1\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [8]{1-\frac{1}{a x}}}{\sqrt [8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 6171
Rule 50
Rule 63
Rule 331
Rule 299
Rule 1122
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{\sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}-\frac{1}{4} \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 \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+(2 a) \operatorname{Subst}\left (\int \frac{x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-\frac{1}{a x}}\right )\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+(2 a) \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 )\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+\frac{a \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 )}{\sqrt{2}}-\frac{a \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 )}{\sqrt{2}}\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}-\frac{a \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 )}{\sqrt{2}}+\frac{a \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 )}{\sqrt{2}}\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+\frac{a \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 )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}+\frac{a \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 )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{a \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 )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}}-\frac{a \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 )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}}\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+\frac{1}{4} \left (\sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} a\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}{4} \left (\sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} a\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}{8} \left (\sqrt{2-\sqrt{2}} a\right ) \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}{8} \left (\sqrt{2-\sqrt{2}} a\right ) \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}{8} \left (\sqrt{2+\sqrt{2}} a\right ) \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}{8} \left (\sqrt{2+\sqrt{2}} a\right ) \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}{4} \left (\sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} a\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}{4} \left (\sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} a\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 )\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \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}{8} \sqrt{2-\sqrt{2}} a \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}{8} \sqrt{2+\sqrt{2}} a \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}{8} \sqrt{2+\sqrt{2}} a \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} \left (\sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} a\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 )-\frac{1}{2} \left (\sqrt{\frac{1}{2} \left (3-2 \sqrt{2}\right )} a\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 )-\frac{1}{2} \left (\sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} a\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 )-\frac{1}{2} \left (\sqrt{\frac{1}{2} \left (3+2 \sqrt{2}\right )} a\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 )\\ &=a \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}}-\frac{1}{4} \sqrt{2+\sqrt{2}} a \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 )-\frac{1}{4} \sqrt{2-\sqrt{2}} a \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 )+\frac{1}{4} \sqrt{2+\sqrt{2}} a \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 )+\frac{1}{4} \sqrt{2-\sqrt{2}} a \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 )+\frac{1}{8} \sqrt{2-\sqrt{2}} a \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}{8} \sqrt{2-\sqrt{2}} a \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}{8} \sqrt{2+\sqrt{2}} a \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}{8} \sqrt{2+\sqrt{2}} a \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 )\\ \end{align*}
Mathematica [C] time = 0.0608243, size = 46, normalized size = 0.07 \[ -2 a e^{\frac{1}{4} \coth ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},-e^{2 \coth ^{-1}(a x)}\right )-\frac{1}{e^{2 \coth ^{-1}(a x)}+1}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\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^{2} \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 = 2.19607, size = 7880, normalized size = 11.66 \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 = 1.2168, size = 583, normalized size = 0.86 \begin{align*} \frac{1}{8} \,{\left (2 \, \sqrt{-\sqrt{2} + 2} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2}}\right ) + 2 \, \sqrt{-\sqrt{2} + 2} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2}}\right ) + 2 \, \sqrt{\sqrt{2} + 2} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2}}\right ) + 2 \, \sqrt{\sqrt{2} + 2} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2}}\right ) - \sqrt{\sqrt{2} + 2} \log \left (\sqrt{\sqrt{2} + 2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \sqrt{\sqrt{2} + 2} \log \left (-\sqrt{\sqrt{2} + 2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - \sqrt{-\sqrt{2} + 2} \log \left (\sqrt{-\sqrt{2} + 2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \sqrt{-\sqrt{2} + 2} \log \left (-\sqrt{-\sqrt{2} + 2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \frac{16 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{\frac{a x - 1}{a x + 1} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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