Optimal. Leaf size=392 \[ -\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{7/8} \left (\frac{1}{a x}+1\right )^{9/8}+\frac{x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a} \]
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Rubi [A] time = 0.247184, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167, Rules used = {6171, 96, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}+\frac{\log \left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{32 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}+1\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{\frac{1}{a x}+1}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{1}{2} x^2 \left (1-\frac{1}{a x}\right )^{7/8} \left (\frac{1}{a x}+1\right )^{9/8}+\frac{x \left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{\frac{1}{a x}+1}}{8 a} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 96
Rule 94
Rule 93
Rule 214
Rule 212
Rule 206
Rule 203
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{\frac{1}{4} \coth ^{-1}(a x)} x \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^3 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [8]{1+\frac{x}{a}}}{x^2 \sqrt [8]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\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 )}{32 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\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 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\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 a^2}+\frac{\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 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}+\frac{\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 )}{16 a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\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 )}{32 a^2}+\frac{\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 )}{32 a^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 )}{32 \sqrt{2} a^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 )}{32 \sqrt{2} a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^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 )}{32 \sqrt{2} a^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 )}{32 \sqrt{2} a^2}+\frac{\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 )}{16 \sqrt{2} a^2}-\frac{\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 )}{16 \sqrt{2} a^2}\\ &=\frac{\left (1-\frac{1}{a x}\right )^{7/8} \sqrt [8]{1+\frac{1}{a x}} x}{8 a}+\frac{1}{2} \left (1-\frac{1}{a x}\right )^{7/8} \left (1+\frac{1}{a x}\right )^{9/8} x^2-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{1+\frac{1}{a x}}}{\sqrt [8]{1-\frac{1}{a x}}}\right )}{16 a^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 )}{32 \sqrt{2} a^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 )}{32 \sqrt{2} a^2}\\ \end{align*}
Mathematica [A] time = 0.724106, size = 319, normalized size = 0.81 \[ \frac{\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}-1}+\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}+1}-\frac{12 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\frac{1}{2} \coth ^{-1}(a x)}+1}-\frac{40 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\coth ^{-1}(a x)}+1}+\frac{2}{\left (e^{\frac{1}{4} \coth ^{-1}(a x)}-1\right )^2}-\frac{2}{\left (e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )^2}+\frac{8 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )^2}+\frac{32 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left (e^{\coth ^{-1}(a x)}+1\right )^2}-\sqrt{2} \log \left (-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+\sqrt{2} \log \left (\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+4 \tan ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right )+4 \tanh ^{-1}\left (e^{\frac{1}{4} \coth ^{-1}(a x)}\right )}{64 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.139, size = 0, normalized size = 0. \begin{align*} \int{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 [A] time = 1.55323, size = 410, normalized size = 1.05 \begin{align*} \frac{1}{64} \, a{\left (\frac{16 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{15}{8}} - 9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{3}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right ) - \sqrt{2} \log \left (\sqrt{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{2} \log \left (-\sqrt{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 )}{a^{3}} - \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{3}} + \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{3}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73388, size = 1269, normalized size = 3.24 \begin{align*} \frac{4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{1}{4}} - 1\right ) + 4 \, \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \arctan \left (\sqrt{2} \sqrt{-\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} - \sqrt{2} a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{1}{4}} + 1\right ) + \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - \sqrt{2} a^{2} \frac{1}{a^{8}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} \frac{1}{a^{8}}^{\frac{3}{4}} + a^{4} \sqrt{\frac{1}{a^{8}}} + \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 8 \,{\left (4 \, a^{2} x^{2} + 9 \, a x + 5\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}} - 4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right ) + 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right ) - 2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1\right )}{64 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt [8]{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20437, size = 405, normalized size = 1.03 \begin{align*} -\frac{1}{64} \, a{\left (\frac{2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{3}} + \frac{2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}\right )}{a^{3}} - \frac{\sqrt{2} \log \left (\sqrt{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 )}{a^{3}} + \frac{\sqrt{2} \log \left (-\sqrt{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 )}{a^{3}} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}}\right )}{a^{3}} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} + 1\right )}{a^{3}} + \frac{2 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{8}} - 1 \right |}\right )}{a^{3}} + \frac{16 \,{\left (\frac{{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}}{a x + 1} - 9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{8}}\right )}}{a^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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