Integrand size = 12, antiderivative size = 20 \[ \int e^{2 i \arctan (a+b x)} \, dx=-x+\frac {2 i \log (i+a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5201, 45} \[ \int e^{2 i \arctan (a+b x)} \, dx=-x+\frac {2 i \log (a+b x+i)}{b} \]
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Rule 45
Rule 5201
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a+i b x}{1-i a-i b x} \, dx \\ & = \int \left (-1+\frac {2 i}{i+a+b x}\right ) \, dx \\ & = -x+\frac {2 i \log (i+a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int e^{2 i \arctan (a+b x)} \, dx=-x+\frac {2 \arctan (a+b x)}{b}+\frac {i \log \left (1+(a+b x)^2\right )}{b} \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(\frac {2 i \ln \left (b x +a +i\right )-b x}{b}\) | \(21\) |
risch | \(-x +\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {2 \arctan \left (b x +a \right )}{b}\) | \(40\) |
default | \(-x +\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b}+\frac {2 \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\) | \(51\) |
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{2 i \arctan (a+b x)} \, dx=-\frac {b x - 2 i \, \log \left (\frac {b x + a + i}{b}\right )}{b} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int e^{2 i \arctan (a+b x)} \, dx=- x + \frac {2 i \log {\left (a + b x + i \right )}}{b} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int e^{2 i \arctan (a+b x)} \, dx=-x + \frac {2 \, \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b} + \frac {i \, \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b} \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int e^{2 i \arctan (a+b x)} \, dx=-x + \frac {2 i \, \log \left (b x + a + i\right )}{b} \]
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Time = 0.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int e^{2 i \arctan (a+b x)} \, dx=-x+\frac {\ln \left (x+\frac {a+1{}\mathrm {i}}{b}\right )\,2{}\mathrm {i}}{b} \]
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