Integrand size = 12, antiderivative size = 176 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=-\frac {25 \sqrt [4]{1+\frac {1}{a x}}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {25 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2} \]
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Time = 0.05 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6306, 98, 96, 95, 218, 212, 209} \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=\frac {25 \arctan \left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \text {arctanh}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}-\frac {25 \sqrt [4]{\frac {1}{a x}+1}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {x^2 \left (\frac {1}{a x}+1\right )^{9/4}}{2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 x \left (\frac {1}{a x}+1\right )^{5/4}}{4 a \sqrt [4]{1-\frac {1}{a x}}} \]
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Rule 95
Rule 96
Rule 98
Rule 209
Rule 212
Rule 218
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{x^3 \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}-\frac {5 \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/4}}{x^2 \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{4 a} \\ & = \frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}-\frac {25 \text {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{x \left (1-\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = -\frac {25 \sqrt [4]{1+\frac {1}{a x}}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}-\frac {25 \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = -\frac {25 \sqrt [4]{1+\frac {1}{a x}}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}-\frac {25 \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 )}{2 a^2} \\ & = -\frac {25 \sqrt [4]{1+\frac {1}{a x}}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {25 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2} \\ & = -\frac {25 \sqrt [4]{1+\frac {1}{a x}}}{2 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {5 \left (1+\frac {1}{a x}\right )^{5/4} x}{4 a \sqrt [4]{1-\frac {1}{a x}}}+\frac {\left (1+\frac {1}{a x}\right )^{9/4} x^2}{2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {25 \arctan \left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2}+\frac {25 \text {arctanh}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{4 a^2} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.45 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=\frac {-\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)} \left (25-45 e^{2 \coth ^{-1}(a x)}+16 e^{4 \coth ^{-1}(a x)}\right )}{\left (-1+e^{2 \coth ^{-1}(a x)}\right )^2}+25 \arctan \left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+25 \text {arctanh}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{4 a^2} \]
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\[\int \frac {x}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}d x\]
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Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.73 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=-\frac {50 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 25 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 25 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (2 \, a^{3} x^{3} + 11 \, a^{2} x^{2} - 34 \, a x - 43\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{8 \, {\left (a^{3} x - a^{2}\right )}} \]
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\[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=\int \frac {x}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=\frac {1}{8} \, a {\left (\frac {4 \, {\left (\frac {45 \, {\left (a x - 1\right )}}{a x + 1} - \frac {25 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 16\right )}}{a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 2 \, a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {50 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} + \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} - \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{3}}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.91 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=-\frac {1}{8} \, a {\left (\frac {50 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{3}} - \frac {25 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{3}} + \frac {25 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{3}} + \frac {64}{a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {4 \, {\left (\frac {9 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - 13 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
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Time = 4.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79 \[ \int e^{\frac {5}{2} \coth ^{-1}(a x)} x \, dx=\frac {25\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}-\frac {25\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{4\,a^2}-\frac {\frac {25\,{\left (a\,x-1\right )}^2}{2\,{\left (a\,x+1\right )}^2}-\frac {45\,\left (a\,x-1\right )}{2\,\left (a\,x+1\right )}+8}{a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-2\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}} \]
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