Integrand size = 14, antiderivative size = 356 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}-\frac {3 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \]
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Time = 0.25 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6306, 92, 81, 52, 65, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {3 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/4}-\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}+\frac {3 a^3 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{16 \sqrt {2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (\frac {1}{a x}+1\right )^{3/4}}{3 x} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}+\frac {1}{3} a^2 \text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}} \left (-1+\frac {x}{2 a}\right )}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}-\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}-\frac {1}{16} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}+\frac {1}{4} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right ) \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}+\frac {1}{4} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}+\frac {1}{16} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {1}{16} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}-\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \\ & = -\frac {3}{8} a^3 \sqrt [4]{1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/4}-\frac {1}{12} a^3 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/4} \left (1+\frac {1}{a x}\right )^{3/4}}{3 x}-\frac {3 a^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {3 a^3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}}+\frac {3 a^3 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.26 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} a^3 \left (-\frac {8 e^{\frac {3}{2} \coth ^{-1}(a x)} \left (29+6 e^{2 \coth ^{-1}(a x)}+9 e^{4 \coth ^{-1}(a x)}\right )}{\left (1+e^{2 \coth ^{-1}(a x)}\right )^3}+9 \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\coth ^{-1}(a x)+2 \log \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ]\right ) \]
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\[\int \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}}{x^{4}}d x\]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.61 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {9 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (3 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 3 \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) + 9 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (3 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 3 i \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 9 i \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (3 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 3 i \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 9 \, \left (-a^{12}\right )^{\frac {1}{4}} x^{3} \log \left (3 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 3 \, \left (-a^{12}\right )^{\frac {1}{4}}\right ) - 2 \, {\left (11 \, a^{3} x^{3} + a^{2} x^{2} - 2 \, a x + 8\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{48 \, x^{3}} \]
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\[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {\sqrt [4]{\frac {a x - 1}{a x + 1}}}{x^{4}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} \, {\left (18 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 18 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 9 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, {\left (29 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 6 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
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Time = 0.31 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{96} \, {\left (18 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 18 \, \sqrt {2} a^{2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 9 \, \sqrt {2} a^{2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \frac {8 \, {\left (\frac {6 \, {\left (a x - 1\right )} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + \frac {29 \, {\left (a x - 1\right )}^{2} a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + 9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \]
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Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-\frac {1}{2} \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {\frac {3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{4}+\frac {a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}+\frac {29\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{12}}{\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {3\,\left (a\,x-1\right )}{a\,x+1}+1}-\frac {{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,3{}\mathrm {i}}{8}-\frac {{\left (-1\right )}^{1/4}\,a^3\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )\,3{}\mathrm {i}}{8} \]
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