Optimal. Leaf size=18 \[ -\frac {e^{-2 \text {ArcTan}(a x)}}{2 a c} \]
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Rubi [A]
time = 0.02, 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 = {5179}
\begin {gather*} -\frac {e^{-2 \text {ArcTan}(a x)}}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 5179
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] Result contains complex when optimal does not.
time = 0.01, size = 34, normalized size = 1.89 \begin {gather*} -\frac {(1-i a x)^{-i} (1+i a x)^i}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 18, normalized size = 1.00
method | result | size |
gosper | \(-\frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{2 a c}\) | \(18\) |
risch | \(-\frac {\left (-i a x +1\right )^{-i} \left (i a x +1\right )^{i}}{2 a c}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 23, normalized size = 1.28 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.82, size = 15, normalized size = 0.83 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.60, size = 19, normalized size = 1.06 \begin {gather*} \begin {cases} - \frac {e^{- 2 \operatorname {atan}{\left (a x \right )}}}{2 a c} & \text {for}\: a \neq 0 \\\frac {x}{c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 15, normalized size = 0.83 \begin {gather*} -\frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end {gather*}
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 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{2\,a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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