Integrand size = 24, antiderivative size = 34 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \]
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Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3568} \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \]
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Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}(\int (a-x) \, dx,x,i a \tan (c+d x))}{a^3 d} \\ & = \frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\tan (c+d x)}{a d}-\frac {i \tan ^2(c+d x)}{2 a d} \]
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Time = 1.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}\) | \(23\) |
derivativedivides | \(-\frac {i \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+i \tan \left (d x +c \right )\right )}{a d}\) | \(30\) |
default | \(-\frac {i \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+i \tan \left (d x +c \right )\right )}{a d}\) | \(30\) |
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none
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 i}{a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d} \]
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\[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\sec ^{4}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
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none
Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \]
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none
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i \, \tan \left (d x + c\right )^{2} - 2 \, \tan \left (d x + c\right )}{2 \, a d} \]
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Time = 3.90 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^4(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,d} \]
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