Integrand size = 28, antiderivative size = 80 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3583, 3856, 2719} \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d (a+i a \tan (c+d x)) \sqrt {e \sec (c+d x)}} \]
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Rule 2719
Rule 3583
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a} \\ & = \frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \sqrt {\cos (c+d x)} \, dx}{5 a \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {\left (4+4 \cos (2 (c+d x))-2 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+3 i \sin (2 (c+d x))\right ) (i+\tan (c+d x))}{5 a d \sqrt {e \sec (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (94 ) = 188\).
Time = 7.54 (sec) , antiderivative size = 440, normalized size of antiderivative = 5.50
method | result | size |
default | \(-\frac {2 i \left (i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-3 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+i \cos \left (d x +c \right ) \sin \left (d x +c \right )-\left (\cos ^{3}\left (d x +c \right )\right )+6 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+3 i \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )+3 \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )}{5 a d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}}\) | \(440\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 12 i \, \sqrt {2} \sqrt {e} e^{\left (3 i \, d x + 3 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{10 \, a d e} \]
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\[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )} - i \sqrt {e \sec {\left (c + d x \right )}}}\, dx}{a} \]
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Exception generated. \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\int { \frac {1}{\sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx=\int \frac {1}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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