Optimal. Leaf size=46 \[ -\frac {2}{\sqrt {\frac {1-a x}{a x+1}}+1}-2 \tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
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Rubi [A] time = 0.40, antiderivative size = 46, 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, 801, 203} \[ -\frac {2}{\sqrt {\frac {1-a x}{a x+1}}+1}-2 \tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 801
Rule 6337
Rubi steps
\begin {align*} \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx &=\int \frac {1}{x \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\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x}{(1+x)^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (-\frac {1}{2 (1+x)^2}+\frac {1}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\frac {2}{1+\sqrt {\frac {1-a x}{1+a x}}}-2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=-\frac {2}{1+\sqrt {\frac {1-a x}{1+a x}}}-2 \tan ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 74, normalized size = 1.61 \[ \sqrt {\frac {1-a x}{a x+1}} \left (\frac {1}{a x}+1\right )-\frac {1}{a x}+i \log \left (2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)-2 i a x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.70, size = 76, normalized size = 1.65 \[ \frac {a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - a x \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right ) - 1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right ) x}\, dx \]
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 \[ \int \frac {1}{x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.85, size = 184, normalized size = 4.00 \[ \ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}-\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}-\frac {1}{a\,x}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2\,\left (1+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}\right )} \]
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 \[ a \int \frac {1}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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