Integrand size = 14, antiderivative size = 351 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {25 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}} \]
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Time = 0.22 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6306, 79, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {25 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{4 \sqrt {2}}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (\frac {1}{a x}+1\right )^{5/4}-\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}}+\frac {25 a^2 \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{8 \sqrt {2}} \]
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Rule 52
Rule 65
Rule 79
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \left (1+\frac {x}{a}\right )^{5/4}}{\left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+(5 a) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{4} (25 a) \text {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{8} (25 a) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \left (25 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right ) \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} \left (25 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right ) \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {1}{4} \left (25 a^2\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}{4} \left (25 a^2\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 {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{8} \left (25 a^2\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}{8} \left (25 a^2\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 (25 a^2\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 )}{8 \sqrt {2}}-\frac {\left (25 a^2\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 )}{8 \sqrt {2}} \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {25 a^2 \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 )}{8 \sqrt {2}}-\frac {\left (25 a^2\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 )}{4 \sqrt {2}}+\frac {\left (25 a^2\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 )}{4 \sqrt {2}} \\ & = -\frac {25}{4} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {5}{2} a^2 \left (1-\frac {1}{a x}\right )^{3/4} \left (1+\frac {1}{a x}\right )^{5/4}-\frac {2 a^2 \left (1+\frac {1}{a x}\right )^{9/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {25 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{4 \sqrt {2}}-\frac {25 a^2 \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 )}{8 \sqrt {2}}+\frac {25 a^2 \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 )}{8 \sqrt {2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{16} a^2 \left (-128 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {32 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (1+e^{2 \coth ^{-1}(a x)}\right )^2}-\frac {104 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-50 \sqrt {2} \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+50 \sqrt {2} \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )-25 \sqrt {2} \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )+25 \sqrt {2} \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )\right ) \]
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\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{3}}d x\]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (a x^{3} - x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 15625 \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (i \, a x^{3} - i \, x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 15625 i \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (-i \, a x^{3} + i \, x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 15625 i \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) - 25 \, \left (-a^{8}\right )^{\frac {1}{4}} {\left (a x^{3} - x^{2}\right )} \log \left (15625 \, a^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 15625 \, \left (-a^{8}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (43 \, a^{3} x^{3} + 34 \, a^{2} x^{2} - 11 \, a x - 2\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{8 \, {\left (a x^{3} - x^{2}\right )}} \]
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\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{16} \, {\left (25 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right )\right )} a + \frac {8 \, {\left (\frac {45 \, {\left (a x - 1\right )} a}{a x + 1} + \frac {25 \, {\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 16 \, a\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
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Time = 0.32 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.69 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{16} \, {\left (50 \, \sqrt {2} a \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 50 \, \sqrt {2} a \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 25 \, \sqrt {2} a \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 25 \, \sqrt {2} a \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {128 \, a}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {8 \, {\left (\frac {9 \, {\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} + 13 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]
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Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^3} \, dx=\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4}-\frac {25\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4}-\frac {8\,a^2+\frac {25\,a^2\,{\left (a\,x-1\right )}^2}{2\,{\left (a\,x+1\right )}^2}+\frac {45\,a^2\,\left (a\,x-1\right )}{2\,\left (a\,x+1\right )}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}} \]
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