Optimal. Leaf size=16 \[ -\frac {e^{-\tanh ^{-1}(a x)}}{a c} \]
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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 = {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 = 27, normalized size = 1.69 \[ -\frac {\sqrt {1-a x}}{a c \sqrt {a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 31, normalized size = 1.94 \[ -\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 37, normalized size = 2.31 \[ \frac {2}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 1.75 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a c \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 46, normalized size = 2.88 \[ \frac {\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {1}{a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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