Integrand size = 26, antiderivative size = 29 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 32} \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rule 32
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {1}{(a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = \frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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Time = 1.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 i}{3 a d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(24\) |
default | \(\frac {2 i}{3 a d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(24\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{6 \, a^{3} d} \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2 i}{3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a d} \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 4.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {2{}\mathrm {i}}{3\,a\,d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \]
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