Integrand size = 22, antiderivative size = 27 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {i (a+i a \tan (c+d x))^2}{2 a d} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3567, 3852, 8} \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^2(c+d x)}{2 d} \]
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Rule 8
Rule 3567
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {i a \sec ^2(c+d x)}{2 d}+a \int \sec ^2(c+d x) \, dx \\ & = \frac {i a \sec ^2(c+d x)}{2 d}-\frac {a \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {i a \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {i a \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.62 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(-\frac {i a \left (-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+i \tan \left (d x +c \right )\right )}{d}\) | \(28\) |
default | \(-\frac {i a \left (-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+i \tan \left (d x +c \right )\right )}{d}\) | \(28\) |
risch | \(\frac {2 i a \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}\) | \(34\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2 \, {\left (-2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
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Time = 0.74 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=\begin {cases} \frac {\frac {i a \tan ^{2}{\left (c + d x \right )}}{2} + a \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\left (c \right )} + a\right ) \sec ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \]
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none
Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {-i \, a \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{2 \, d} \]
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Time = 3.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,d} \]
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