Integrand size = 12, antiderivative size = 91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )}+\frac {9}{2} a^2 \csc ^{-1}(a x) \]
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Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6304, 1647, 1607, 12, 807, 679, 222} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {9}{2} a^2 \csc ^{-1}(a x)-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )} \]
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Rule 12
Rule 222
Rule 679
Rule 807
Rule 1607
Rule 1647
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \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 {\left (-a x-x^2\right ) \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-x) x \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 \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 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}+(3 a) \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )}+\frac {1}{2} (9 a) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{1-\frac {x}{a}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )}+\frac {1}{2} (9 a) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )}+\frac {9}{2} a^2 \csc ^{-1}(a x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (1+5 a x-14 a^2 x^2\right )}{x (-1+a x)}+9 a \arcsin \left (\frac {1}{a x}\right )\right ) \]
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Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.37
method | result | size |
risch | \(-\frac {\left (a x -1\right ) \left (6 a x +1\right )}{2 x^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {9 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(125\) |
default | \(\frac {-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-11 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-18 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-4 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+12 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+4 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+9 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}}{2 \sqrt {a^{2}}\, x^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(641\) |
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Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-\frac {18 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (14 \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 6 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{3} - x^{2}\right )}} \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\int \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=-{\left (9 \, a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {15 \, {\left (a x - 1\right )} a}{a x + 1} + \frac {9 \, {\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 4 \, a}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \]
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\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\int { \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
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Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^3} \, dx=\frac {1}{2\,x^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-\frac {7\,a^2}{\sqrt {\frac {a\,x-1}{a\,x+1}}}-9\,a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {5\,a}{2\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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