Optimal. Leaf size=18 \[ -\frac {e^{-2 \tan ^{-1}(a x)}}{2 a c} \]
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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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5071} \[ -\frac {e^{-2 \tan ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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Rule 5071
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)}}{2 a c}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 1.89 \[ -\frac {(1-i a x)^{-i} (1+i a x)^i}{2 a c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 15, normalized size = 0.83 \[ -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 15, normalized size = 0.83 \[ -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 1.00 \[ -\frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{2 a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 23, normalized size = 1.28 \[ -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 15, normalized size = 0.83 \[ -\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{2\,a\,c} \]
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 \[ \begin {cases} - \frac {e^{- 2 \operatorname {atan}{\left (a x \right )}}}{2 a c} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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