Optimal. Leaf size=16 \[ -\frac {e^{-\tan ^{-1}(a x)}}{a c} \]
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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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5071} \[ -\frac {e^{-\tan ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 5071
Rubi steps
\begin {align*} \int \frac {e^{-\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac {e^{-\tan ^{-1}(a x)}}{a c}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 2.25 \[ -\frac {(1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}}}{a c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 15, normalized size = 0.94 \[ -\frac {e^{\left (-\arctan \left (a x\right )\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 15, normalized size = 0.94 \[ -\frac {e^{\left (-\arctan \left (a x\right )\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 1.00 \[ -\frac {{\mathrm e}^{-\arctan \left (a x \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 23, normalized size = 1.44 \[ -\frac {2 \, e^{\left (-\arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 15, normalized size = 0.94 \[ -\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{a\,c} \]
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 \[ \begin {cases} - \frac {e^{- \operatorname {atan}{\left (a x \right )}}}{a c} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{- \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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