Optimal. Leaf size=72 \[ \frac {a}{\sqrt {\frac {1-a x}{a x+1}}+1}-\frac {a}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+a \left (-\tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )\right ) \]
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Rubi [A] time = 0.38, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6337, 77, 207} \[ \frac {a}{\sqrt {\frac {1-a x}{a x+1}}+1}-\frac {a}{\left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}+a \left (-\tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 77
Rule 207
Rule 6337
Rubi steps
\begin {align*} \int \frac {e^{-\text {sech}^{-1}(a x)}}{x^2} \, dx &=\int \frac {1}{x^2 \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 )} \, dx\\ &=(4 a) \operatorname {Subst}\left (\int \frac {x}{(-1+x) (1+x)^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=(4 a) \operatorname {Subst}\left (\int \left (\frac {1}{2 (1+x)^3}-\frac {1}{4 (1+x)^2}+\frac {1}{4 \left (-1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a}{1+\sqrt {\frac {1-a x}{1+a x}}}+a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {a}{\left (1+\sqrt {\frac {1-a x}{1+a x}}\right )^2}+\frac {a}{1+\sqrt {\frac {1-a x}{1+a x}}}-a \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 92, normalized size = 1.28 \[ \frac {1}{2} \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x^2}-\frac {1}{a x^2}+a \log (x)-a \log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 128, normalized size = 1.78 \[ -\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 2}{4 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 110, normalized size = 1.53 \[ -\frac {1}{2} \, {\left (\sqrt {a^{2} + \frac {a}{x}} \sqrt {-a^{2} + \frac {a}{x}} {\left (\frac {1}{a^{2}} - \frac {a^{2} + \frac {a}{x}}{a^{4}}\right )} - \frac {2 \, {\left (a^{2} + \frac {a}{x}\right )} a^{2} - {\left (a^{2} + \frac {a}{x}\right )}^{2}}{a^{4}} - 2 \, \log \left (\sqrt {a^{2} + \frac {a}{x}} - \sqrt {-a^{2} + \frac {a}{x}}\right )\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.05, size = 323, normalized size = 4.49 \[ 2\,a\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )-a\,\mathrm {acosh}\left (\frac {1}{a\,x}\right )-\frac {1}{2\,a\,x^2}-\frac {a\,\left (\frac {14\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}+\frac {14\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^5}+\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^7}+\frac {2\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{1+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {1}{a x^{2} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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