Integrand size = 24, antiderivative size = 23 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
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Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 31} \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \]
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Rule 31
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = \frac {x}{a}+\frac {i \log (\cos (c+d x))}{a d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i \log (i-\tan (c+d x))}{a d} \]
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Time = 0.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(-\frac {i \ln \left (a +i a \tan \left (d x +c \right )\right )}{a d}\) | \(23\) |
default | \(-\frac {i \ln \left (a +i a \tan \left (d x +c \right )\right )}{a d}\) | \(23\) |
risch | \(\frac {2 x}{a}+\frac {2 c}{a d}+\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a d}\) | \(38\) |
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none
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 \, d x + i \, \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{a d} \]
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\[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\sec ^{2}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
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none
Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i \, \log \left (i \, a \tan \left (d x + c\right ) + a\right )}{a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (21) = 42\).
Time = 0.36 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.48 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {-\frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a} + \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a} - \frac {i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a}}{d} \]
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Time = 4.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a\,d} \]
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