Integrand size = 30, antiderivative size = 36 \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i \sqrt {e \sec (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {3569} \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i \sqrt {e \sec (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \]
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Rule 3569
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \sqrt {e \sec (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i \sqrt {e \sec (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \]
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Time = 11.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {2 i \sqrt {e \sec \left (d x +c \right )}}{d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}}\) | \(32\) |
risch | \(\frac {2 i \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}{\sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(59\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \, \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{a d} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 i \, \sqrt {e} \sqrt {-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}}{\sqrt {a} d \sqrt {-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {\sqrt {e \sec \left (d x + c\right )}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Time = 5.70 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,2{}\mathrm {i}}{d\,\sqrt {a+\frac {a\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}}{\cos \left (c+d\,x\right )}}} \]
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