Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
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Rubi [A] time = 0.10, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6175, 6178, 266, 63, 208} \[ -\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx &=-\frac {\int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right ) x} \, dx}{a c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c}\\ &=-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.48 \[ -\frac {\log \left (a x \left (\sqrt {\frac {a^2 x^2-1}{a^2 x^2}}+1\right )\right )}{a c} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.49, size = 47, normalized size = 2.04 \[ -\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 33, normalized size = 1.43 \[ \frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 76, normalized size = 3.30 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 55, normalized size = 2.39 \[ -a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 24, normalized size = 1.04 \[ -\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \]
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 \[ - \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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