Optimal. Leaf size=13 \[ \frac{e^{\tanh ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0287232, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6137} \[ \frac{e^{\tanh ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 6137
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{\tanh ^{-1}(a x)}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0077084, size = 26, normalized size = 2. \[ \frac{\sqrt{a x+1}}{a c \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 25, normalized size = 1.9 \begin{align*}{\frac{ax+1}{ac}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4996, size = 220, normalized size = 16.92 \begin{align*} -\frac{a^{2} c{\left (\frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{2} c x + a^{2} c^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} c}{\sqrt{a^{2} c^{2}} a^{2} c x - a^{2} c^{2}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c x + \sqrt{a^{2} c^{2}} a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} c x - \sqrt{a^{2} c^{2}} a}\right )}}{2 \, \sqrt{a^{2} c^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53849, size = 65, normalized size = 5. \begin{align*} \frac{a x - \sqrt{-a^{2} x^{2} + 1} - 1}{a^{2} c x - a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18228, size = 50, normalized size = 3.85 \begin{align*} \frac{2}{c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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