Integrand size = 20, antiderivative size = 19 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=i a x+\frac {a \log (\sin (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3612, 3556} \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \log (\sin (c+d x))}{d}+i a x \]
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Rule 3556
Rule 3612
Rubi steps \begin{align*} \text {integral}& = i a x+a \int \cot (c+d x) \, dx \\ & = i a x+\frac {a \log (\sin (c+d x))}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=i a x+\frac {a \log (\cos (c+d x))}{d}+\frac {a \log (\tan (c+d x))}{d} \]
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Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42
method | result | size |
risch | \(-\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(27\) |
parallelrisch | \(\frac {a \left (2 i d x +2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (\sec ^{2}\left (d x +c \right )\right )\right )}{2 d}\) | \(33\) |
norman | \(i a x +\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(36\) |
derivativedivides | \(\frac {a \left (\ln \left (\tan \left (d x +c \right )\right )-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(37\) |
default | \(\frac {a \left (\ln \left (\tan \left (d x +c \right )\right )-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(37\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {-2 i \, {\left (d x + c\right )} a + a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
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Time = 4.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]
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