Integrand size = 14, antiderivative size = 299 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {5 a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}} \]
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Time = 0.21 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6306, 49, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\frac {5 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}-\frac {5 a \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}}-\frac {4 a \left (\frac {1}{a x}+1\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}-\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}+\frac {5 a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}} \]
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Rule 49
Rule 52
Rule 65
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 {\left (1+\frac {x}{a}\right )^{5/4}}{\left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+5 \text {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right ) \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5}{2} \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 ) \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-(10 a) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right ) \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-(10 a) \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 ) \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+(5 a) \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 )-(5 a) \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 ) \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {1}{2} (5 a) \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}{2} (5 a) \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 {(5 a) \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 )}{2 \sqrt {2}}-\frac {(5 a) \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 )}{2 \sqrt {2}} \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}-\frac {5 a \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 )}{2 \sqrt {2}}+\frac {5 a \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 )}{2 \sqrt {2}}-\frac {(5 a) \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 )}{\sqrt {2}}+\frac {(5 a) \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 )}{\sqrt {2}} \\ & = -5 a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {4 a \left (1+\frac {1}{a x}\right )^{5/4}}{\sqrt [4]{1-\frac {1}{a x}}}+\frac {5 a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {5 a \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 )}{2 \sqrt {2}}+\frac {5 a \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 )}{2 \sqrt {2}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=a \left (-\frac {10 e^{\frac {1}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {8 e^{\frac {5}{2} \coth ^{-1}(a x)}}{1+e^{2 \coth ^{-1}(a x)}}-\frac {5 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}+\frac {5 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}\right )}{2 \sqrt {2}}\right ) \]
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\[\int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}} x^{2}}d x\]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (a x^{2} - x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 125 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (i \, a x^{2} - i \, x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 125 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (-i \, a x^{2} + i \, x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 125 i \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) - 5 \, \left (-a^{4}\right )^{\frac {1}{4}} {\left (a x^{2} - x\right )} \log \left (125 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 125 \, \left (-a^{4}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (9 \, a^{2} x^{2} + 8 \, a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{2 \, {\left (a x^{2} - x\right )}} \]
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\[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
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Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {1}{4} \, {\left (10 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 10 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - 5 \, \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 5 \, \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, {\left (\frac {5 \, {\left (a x - 1\right )}}{a x + 1} + 4\right )}}{\frac {{\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}\right )} a \]
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Time = 4.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.36 \[ \int \frac {e^{\frac {5}{2} \coth ^{-1}(a x)}}{x^2} \, dx=5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-5\,{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-\frac {8\,a+\frac {10\,a\,\left (a\,x-1\right )}{a\,x+1}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}} \]
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