Integrand size = 12, antiderivative size = 147 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6472, 1626, 213} \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {\frac {1-a x}{a x+1}}\right )+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}-\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4} \]
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Rule 213
Rule 1626
Rule 6472
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2}{x^3} \, dx \\ & = \left (4 a^2\right ) \text {Subst}\left (\int \frac {x \left (1+x^2\right )}{(-1+x)^5 (1+x)} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = \left (4 a^2\right ) \text {Subst}\left (\int \left (\frac {1}{(-1+x)^5}+\frac {3}{2 (-1+x)^4}+\frac {3}{4 (-1+x)^3}+\frac {1}{8 (-1+x)^2}-\frac {1}{8 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = -\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = -\frac {a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}+\frac {2 a^2}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}-\frac {3 a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a^2}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}+\frac {1}{2} a^2 \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\frac {\frac {(1+a x) \left (-2+2 a x-2 \sqrt {\frac {1-a x}{1+a x}}+a^2 x^2 \sqrt {\frac {1-a x}{1+a x}}\right )}{x^4}-a^4 \log (x)+a^4 \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^2} \]
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Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {-\frac {1}{4 x^{4}}+\frac {a^{2}}{2 x^{2}}}{a^{2}}+\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{4} x^{4}+a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \,x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {1}{4 a^{2} x^{4}}\) | \(131\) |
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Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 4 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} - 2 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 4}{8 \, a^{2} x^{4}} \]
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\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\frac {\int \frac {2}{x^{5}}\, dx + \int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{4}}\, dx}{a^{2}} \]
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\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=\int { \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{3}} \,d x } \]
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Time = 46.86 (sec) , antiderivative size = 885, normalized size of antiderivative = 6.02 \[ \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^3} \, dx=a^2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )-\frac {\frac {28\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {28\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {4\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {4\,a^2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}-\frac {\frac {23\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {333\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {671\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {671\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^9}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^9}+\frac {333\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{11}}+\frac {23\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{13}}-\frac {3\,a^2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{15}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{15}}-\frac {3\,a^2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}}{1+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}}+\frac {1}{2\,x^2}-\frac {1}{2\,a^2\,x^4} \]
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