Optimal. Leaf size=13 \[ \frac {e^{\tanh ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 2.00 \[ \frac {\sqrt {a x+1}}{a c \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 33, normalized size = 2.54 \[ \frac {a x - \sqrt {-a^{2} x^{2} + 1} - 1}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 37, normalized size = 2.85 \[ \frac {2}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 1.92 \[ \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}\, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 125, normalized size = 9.62 \[ -\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{3} c^{2} x + a^{2} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{3} c^{2} x - a^{2} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c x + a^{2} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c x - a^{2} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 47, normalized size = 3.62 \[ \frac {\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {a x}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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