Integrand size = 21, antiderivative size = 18 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{2 \arctan (a x)}}{2 a c} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5179} \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{2 \arctan (a x)}}{2 a c} \]
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Rule 5179
Rubi steps \begin{align*} \text {integral}& = \frac {e^{2 \arctan (a x)}}{2 a c} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {(1-i a x)^i (1+i a x)^{-i}}{2 a c} \]
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Time = 0.52 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {{\mathrm e}^{2 \arctan \left (a x \right )}}{2 a c}\) | \(16\) |
parallelrisch | \(\frac {{\mathrm e}^{2 \arctan \left (a x \right )}}{2 a c}\) | \(16\) |
risch | \(\frac {\left (-i a x +1\right )^{i} \left (i a x +1\right )^{-i}}{2 a c}\) | \(29\) |
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]
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Time = 0.42 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\begin {cases} \frac {e^{2 \operatorname {atan}{\left (a x \right )}}}{2 a c} & \text {for}\: a \neq 0 \\\frac {x}{c} & \text {otherwise} \end {cases} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]
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none
Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \]
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Time = 0.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{2\,a\,c} \]
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