Integrand size = 14, antiderivative size = 731 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right ) \]
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Time = 0.49 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6306, 81, 52, 65, 338, 305, 1136, 1183, 648, 632, 210, 642} \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}+\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right ) \]
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Rule 52
Rule 65
Rule 81
Rule 210
Rule 305
Rule 338
Rule 632
Rule 642
Rule 648
Rule 1136
Rule 1183
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \text {Subst}\left (\int \frac {\sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} a \text {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 ) \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{4} a^2 \text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-\frac {1}{a x}}\right ) \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{4} a^2 \text {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 ) \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {a^2 \text {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 )}{8 \sqrt {2}}-\frac {a^2 \text {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 )}{8 \sqrt {2}} \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {a^2 \text {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 )}{8 \sqrt {2}}+\frac {a^2 \text {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 )}{8 \sqrt {2}} \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {a^2 \text {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 )}{16 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {a^2 \text {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 )}{16 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {a^2 \text {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 )}{16 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {a^2 \text {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 )}{16 \sqrt {2 \left (2+\sqrt {2}\right )}} \\ & = \frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{32} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \text {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}{32} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \text {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}{64} \left (\sqrt {2-\sqrt {2}} a^2\right ) \text {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}{64} \left (\sqrt {2-\sqrt {2}} a^2\right ) \text {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}{64} \left (\sqrt {2+\sqrt {2}} a^2\right ) \text {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}{64} \left (\sqrt {2+\sqrt {2}} a^2\right ) \text {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}{32} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \text {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}{32} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \text {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} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{64} \sqrt {2-\sqrt {2}} a^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}{64} \sqrt {2-\sqrt {2}} a^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}{64} \sqrt {2+\sqrt {2}} a^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}{64} \sqrt {2+\sqrt {2}} a^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}{16} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \text {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}{16} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \text {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}{16} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \text {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}{16} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \text {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}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \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}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \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}{32} \sqrt {2+\sqrt {2}} a^2 \arctan \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}{32} \sqrt {2-\sqrt {2}} a^2 \arctan \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}{64} \sqrt {2-\sqrt {2}} a^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}{64} \sqrt {2-\sqrt {2}} a^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}{64} \sqrt {2+\sqrt {2}} a^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}{64} \sqrt {2+\sqrt {2}} a^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 ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.10 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {a^2 e^{\frac {1}{4} \coth ^{-1}(a x)} \left (-1-9 e^{2 \coth ^{-1}(a x)}+\left (1+e^{2 \coth ^{-1}(a x)}\right )^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{2 \coth ^{-1}(a x)}\right )\right )}{4 \left (1+e^{2 \coth ^{-1}(a x)}\right )^2} \]
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\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x^{3}}d x\]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {-\left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) - \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + \left (i - 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (2 \, a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (i + 1\right ) \, \sqrt {2} \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 2 \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (-a^{16}\right )^{\frac {7}{8}}\right ) - 2 i \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + i \, \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 2 i \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - i \, \left (-a^{16}\right )^{\frac {7}{8}}\right ) - 2 \, \left (-a^{16}\right )^{\frac {1}{8}} x^{2} \log \left (a^{14} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - \left (-a^{16}\right )^{\frac {7}{8}}\right ) + 8 \, {\left (5 \, a^{2} x^{2} + 9 \, a x + 4\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{64 \, x^{2}} \]
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\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]
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\[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}} \,d x } \]
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Time = 0.50 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{64} \, {\left (2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) - a \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - a \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \frac {16 \, {\left (a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{8}} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]
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Time = 4.50 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.29 \[ \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{4}+\frac {a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{4}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{16}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16}+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right ) \]
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