Integrand size = 24, antiderivative size = 38 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \]
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Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3623, 3612, 3556} \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d}-2 a^2 x \]
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Rule 3556
Rule 3612
Rule 3623
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (c+d x)}{d}+\int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx \\ & = -2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\left (2 i a^2\right ) \int \cot (c+d x) \, dx \\ & = -2 a^2 x-\frac {a^2 \cot (c+d x)}{d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=a^2 \left (-\frac {\cot (c+d x)}{d}+\frac {2 i \log (\tan (c+d x))}{d}-\frac {2 i \log (i+\tan (c+d x))}{d}\right ) \]
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Time = 0.54 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13
method | result | size |
parallelrisch | \(\frac {a^{2} \left (2 i \ln \left (\tan \left (d x +c \right )\right )-i \ln \left (\sec ^{2}\left (d x +c \right )\right )-2 d x -\cot \left (d x +c \right )\right )}{d}\) | \(43\) |
derivativedivides | \(\frac {-a^{2} \left (d x +c \right )+2 i a^{2} \ln \left (\sin \left (d x +c \right )\right )+a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(49\) |
default | \(\frac {-a^{2} \left (d x +c \right )+2 i a^{2} \ln \left (\sin \left (d x +c \right )\right )+a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(49\) |
risch | \(\frac {4 a^{2} c}{d}-\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(54\) |
norman | \(\frac {-\frac {a^{2}}{d}-2 a^{2} x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(68\) |
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none
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (i \, a^{2} + {\left (-i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]
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Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=- \frac {2 i a^{2}}{d e^{2 i c} e^{2 i d x} - d} + \frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (d x + c\right )} a^{2} + i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {a^{2}}{\tan \left (d x + c\right )}}{d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (36) = 72\).
Time = 0.56 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.24 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {8 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 4 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {-4 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
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Time = 4.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {a^2\,\left (\mathrm {cot}\left (c+d\,x\right )+4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\right )}{d} \]
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