Integrand size = 24, antiderivative size = 32 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]
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Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3569} \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]
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Rule 3569
Rubi steps \begin{align*} \text {integral}& = \frac {i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \sec ^3(c+d x)}{3 d (a+i a \tan (c+d x))^3} \]
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Time = 0.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{3 a^{3} d}\) | \(19\) |
derivativedivides | \(\frac {-\frac {8}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{a^{3} d}\) | \(57\) |
default | \(\frac {-\frac {8}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}}{a^{3} d}\) | \(57\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \, e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{3} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (26) = 52\).
Time = 0.80 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\begin {cases} - \frac {\sec ^{3}{\left (c + d x \right )}}{3 a^{3} d \tan ^{3}{\left (c + d x \right )} - 9 i a^{3} d \tan ^{2}{\left (c + d x \right )} - 9 a^{3} d \tan {\left (c + d x \right )} + 3 i a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{3}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
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none
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {i \, \cos \left (3 \, d x + 3 \, c\right ) + \sin \left (3 \, d x + 3 \, c\right )}{3 \, a^{3} d} \]
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none
Time = 0.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{3 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}} \]
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Time = 3.92 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {2\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,a^3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}+1\right )} \]
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