Optimal. Leaf size=14 \[ \frac {1}{a c (a x+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac {1}{a c (a x+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\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 = 0.02, size = 18, normalized size = 1.29 \[ -\frac {e^{-2 \coth ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 12, normalized size = 0.86 \[ \frac {1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 14, 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 = 0.03, size = 15, normalized size = 1.07 \[ \frac {1}{a c \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 12, normalized size = 0.86 \[ \frac {1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 12, normalized size = 0.86 \[ \frac {1}{a\,\left (c+a\,c\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 10, normalized size = 0.71 \[ \frac {1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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