Optimal. Leaf size=16 \[ -\frac {e^{-\coth ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6183} \[ -\frac {e^{-\coth ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 6183
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac {e^{-\coth ^{-1}(a x)}}{a c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 16, normalized size = 1.00 \[ -\frac {e^{-\coth ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 23, normalized size = 1.44 \[ -\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c} \]
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 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 24, normalized size = 1.50 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 23, normalized size = 1.44 \[ -\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 1.44 \[ -\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c} \]
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 \[ - \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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