Optimal. Leaf size=15 \[ \frac {1}{a c (1-a x)} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 32} \[ \frac {1}{a c (1-a x)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac {\int \frac {1}{(1-a x)^2} \, dx}{c}\\ &=\frac {1}{a c (1-a x)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 18, normalized size = 1.20 \[ \frac {e^{2 \tanh ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 15, normalized size = 1.00 \[ -\frac {1}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 15, normalized size = 1.00 \[ -\frac {1}{{\left (a x - 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 1.07 \[ -\frac {1}{c a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 15, normalized size = 1.00 \[ -\frac {1}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 13, normalized size = 0.87 \[ \frac {1}{a\,\left (c-a\,c\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 12, normalized size = 0.80 \[ - \frac {1}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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