Integrand size = 24, antiderivative size = 52 \[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3570, 212} \[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d} \]
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Rule 212
Rule 3570
Rubi steps \begin{align*} \text {integral}& = \frac {(2 i) \text {Subst}\left (\int \frac {1}{2-a x^2} \, dx,x,\frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}}\right )}{d} \\ & = \frac {i \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35 \[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i e^{i (c+d x)} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )}{d \sqrt {1+e^{2 i (c+d x)}} \sqrt {a+i a \tan (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (41 ) = 82\).
Time = 7.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.35
method | result | size |
default | \(\frac {\left (i \cos \left (d x +c \right )+i-\sin \left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\) | \(122\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (39) = 78\).
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.87 \[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {1}{2} i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {4 \, {\left ({\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{d}\right ) + \frac {1}{2} i \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {4 \, {\left ({\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a d^{2}}} - i\right )} e^{\left (-i \, d x - i \, c\right )}}{d}\right ) \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\sec (c+d x)}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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