Integrand size = 13, antiderivative size = 19 \[ \int (a+i a \tan (c+d x)) \, dx=a x-\frac {i a \log (\cos (c+d x))}{d} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3556} \[ \int (a+i a \tan (c+d x)) \, dx=a x-\frac {i a \log (\cos (c+d x))}{d} \]
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Rule 3556
Rubi steps \begin{align*} \text {integral}& = a x+(i a) \int \tan (c+d x) \, dx \\ & = a x-\frac {i a \log (\cos (c+d x))}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int (a+i a \tan (c+d x)) \, dx=a x-\frac {i a \log (\cos (c+d x))}{d} \]
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Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21
method | result | size |
default | \(a x +\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(23\) |
norman | \(a x +\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(23\) |
parallelrisch | \(a x +\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(23\) |
parts | \(a x +\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(23\) |
derivativedivides | \(\frac {a \left (\frac {i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(28\) |
risch | \(-\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {2 a c}{d}\) | \(28\) |
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none
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int (a+i a \tan (c+d x)) \, dx=-\frac {i \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d} \]
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Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int (a+i a \tan (c+d x)) \, dx=- \frac {i a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int (a+i a \tan (c+d x)) \, dx=a x + \frac {i \, a \log \left (\sec \left (d x + c\right )\right )}{d} \]
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none
Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int (a+i a \tan (c+d x)) \, dx=a x - \frac {i \, a \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \]
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Time = 3.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int (a+i a \tan (c+d x)) \, dx=\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{d} \]
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