Optimal. Leaf size=38 \[ -\frac {(2-a x) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5069} \[ -\frac {(2-a x) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5069
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (2-a x)}{5 a c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.97 \[ \frac {(a x-2) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 44, normalized size = 1.16 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - 2\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{5 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 1.08 \[ \frac {\left (a^{2} x^{2}+1\right ) \left (a x -2\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{5 a \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 35, normalized size = 0.92 \[ \frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {x}{5\,c}-\frac {2}{5\,a\,c}\right )}{\sqrt {c\,a^2\,x^2+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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