Integrand size = 26, antiderivative size = 27 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \]
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Time = 0.07 (sec) , antiderivative size = 27, 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} \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \]
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Rule 32
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = -\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{a d} \]
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Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {2 i \sqrt {a +i a \tan \left (d x +c \right )}}{a d}\) | \(24\) |
default | \(-\frac {2 i \sqrt {a +i a \tan \left (d x +c \right )}}{a d}\) | \(24\) |
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none
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{a d} \]
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\[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \, \sqrt {i \, a \tan \left (d x + c\right ) + a}}{a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (21) = 42\).
Time = 0.60 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {2 i \, \sqrt {\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}}{a d} \]
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Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {\sec ^2(c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}\,2{}\mathrm {i}}{a\,d} \]
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