Optimal. Leaf size=48 \[ 2 \tan ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {2}{1-\sqrt {\frac {1-a x}{a x+1}}} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6334, 97, 12, 41, 216} \[ -\frac {\sqrt {1-a x}}{a x \sqrt {\frac {1}{a x+1}}}-\frac {1}{a x}-\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \sin ^{-1}(a x) \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 41
Rule 97
Rule 216
Rule 6334
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x} \, dx &=-\frac {1}{a x}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {\sqrt {1-a x} \sqrt {1+a x}}{x^2} \, dx}{a}\\ &=-\frac {1}{a x}-\frac {\sqrt {1-a x}}{a x \sqrt {\frac {1}{1+a x}}}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {a^2}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a}\\ &=-\frac {1}{a x}-\frac {\sqrt {1-a x}}{a x \sqrt {\frac {1}{1+a x}}}-\left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=-\frac {1}{a x}-\frac {\sqrt {1-a x}}{a x \sqrt {\frac {1}{1+a x}}}-\left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{a x}-\frac {\sqrt {1-a x}}{a x \sqrt {\frac {1}{1+a x}}}-\sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \sin ^{-1}(a x)\\ \end {align*}
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Mathematica [C] time = 0.05, size = 75, normalized size = 1.56 \[ \sqrt {\frac {1-a x}{a x+1}} \left (-\frac {1}{a x}-1\right )-\frac {1}{a x}-i \log \left (2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)-2 i a x\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 77, normalized size = 1.60 \[ -\frac {a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - a x \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right ) + 1}{a x} \]
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 \[ \int \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 92, normalized size = 1.92 \[ -\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (\arctan \left (\frac {\mathrm {csgn}\relax (a ) a x}{\sqrt {-a^{2} x^{2}+1}}\right ) x a +\sqrt {-a^{2} x^{2}+1}\, \mathrm {csgn}\relax (a )\right ) \mathrm {csgn}\relax (a )}{\sqrt {-a^{2} x^{2}+1}}-\frac {1}{a x} \]
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 \[ \frac {-a \arcsin \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1}}{x}}{a} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 184, normalized size = 3.83 \[ -\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}+\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}-\frac {1}{a\,x}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2\,\left (1+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}\right )} \]
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 \[ \frac {\int \frac {1}{x^{2}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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