Integrand size = 12, antiderivative size = 93 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {11}{2} a^3 \csc ^{-1}(a x) \]
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Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6304, 1647, 1607, 12, 866, 1649, 1668, 794, 222} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {11}{2} a^3 \csc ^{-1}(a x)-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rule 12
Rule 222
Rule 794
Rule 866
Rule 1607
Rule 1647
Rule 1649
Rule 1668
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^2}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (-a x^2-x^3\right )}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x) x^2 \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a} \\ & = -\frac {\text {Subst}\left (\int \frac {a^2 x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{a^2} \\ & = -\text {Subst}\left (\int \frac {x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2 \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{3} \text {Subst}\left (\int \frac {\left (-5-\frac {3 x}{a}\right ) \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {1}{2} \left (11 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {11}{2} a^3 \csc ^{-1}(a x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2+7 a x+19 a^2 x^2-52 a^3 x^3\right )}{x^2 (-1+a x)}+33 a^2 \arcsin \left (\frac {1}{a x}\right )\right ) \]
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Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.45
method | result | size |
risch | \(-\frac {\left (a x -1\right ) \left (28 a^{2} x^{2}+9 a x +2\right )}{6 x^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {11 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a^{2} \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(135\) |
default | \(\frac {-30 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{6} x^{6}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+93 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+33 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, a^{5} x^{5}+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-30 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{5} x^{5}-30 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-51 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-96 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}-66 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-12 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}+60 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+14 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+33 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+33 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-30 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-30 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+5 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}}{6 \sqrt {a^{2}}\, x^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(666\) |
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.03 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {66 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (52 \, a^{4} x^{4} + 33 \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a x^{4} - x^{3}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {75 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {88 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 12 \, a^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=\int { \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 4.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.63 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {4\,a^3+\frac {88\,a^3\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {11\,a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {25\,a^3\,\left (a\,x-1\right )}{a\,x+1}}{\sqrt {\frac {a\,x-1}{a\,x+1}}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}-11\,a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \]
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