Optimal. Leaf size=18 \[ -\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]
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Rubi [A] time = 0.03, antiderivative size = 18, 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} \[ -\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 6137
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac {e^{-3 \tanh ^{-1}(a x)}}{3 a c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.61 \[ -\frac {(1-a x)^{3/2}}{3 a c (a x+1)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.65, size = 55, normalized size = 3.06 \[ -\frac {a^{2} x^{2} + 2 \, a x - \sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )} + 1}{3 \, {\left (a^{3} c x^{2} + 2 \, a^{2} c x + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 66, normalized size = 3.67 \[ \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + 1\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 1.56 \[ -\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a c \left (a x +1\right )^{3}} \]
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 \[ -\int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 32, normalized size = 1.78 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (a\,x-1\right )}{3\,a\,c\,{\left (a\,x+1\right )}^2} \]
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 \[ \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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