Integrand size = 22, antiderivative size = 47 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 \sin (c+d x)}{3 a d}+\frac {i \cos (c+d x)}{3 d (a+i a \tan (c+d x))} \]
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Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3583, 2717} \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 \sin (c+d x)}{3 a d}+\frac {i \cos (c+d x)}{3 d (a+i a \tan (c+d x))} \]
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Rule 2717
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \cos (c+d x)}{3 d (a+i a \tan (c+d x))}+\frac {2 \int \cos (c+d x) \, dx}{3 a} \\ & = \frac {2 \sin (c+d x)}{3 a d}+\frac {i \cos (c+d x)}{3 d (a+i a \tan (c+d x))} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\sec (c+d x) (-3+\cos (2 (c+d x))+2 i \sin (2 (c+d x)))}{6 a d (-i+\tan (c+d x))} \]
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Time = 0.75 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{12 a d}+\frac {i \cos \left (d x +c \right )}{4 a d}+\frac {3 \sin \left (d x +c \right )}{4 a d}\) | \(49\) |
derivativedivides | \(\frac {-\frac {2}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 i}}{a d}\) | \(75\) |
default | \(\frac {-\frac {2}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 i}}{a d}\) | \(75\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}+\frac {2 \tan \left (d x +c \right )}{3 a d}-\frac {2 i \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2 i \left (\tan ^{2}\left (d x +c \right )\right )}{3 a d}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 a d}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (d x +c \right )}{3 a d}+\frac {2 i \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \tan \left (d x +c \right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (1+\tan ^{2}\left (d x +c \right )\right )}\) | \(172\) |
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none
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\left (-3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{12 \, a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (36) = 72\).
Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.68 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\begin {cases} \frac {\left (- 24 i a^{2} d^{2} e^{5 i c} e^{i d x} + 48 i a^{2} d^{2} e^{3 i c} e^{- i d x} + 8 i a^{2} d^{2} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{96 a^{3} d^{3}} & \text {for}\: a^{3} d^{3} e^{4 i c} \neq 0 \\\frac {x \left (e^{4 i c} + 2 e^{2 i c} + 1\right ) e^{- 3 i c}}{4 a} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\frac {3}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}} + \frac {9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}}}{6 \, d} \]
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Time = 3.97 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\left (-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^3} \]
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