Integrand size = 26, antiderivative size = 60 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {2 i a \sqrt {e \sec (c+d x)}}{d}+\frac {2 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d} \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3567, 3856, 2720} \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {2 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a \sqrt {e \sec (c+d x)}}{d} \]
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Rule 2720
Rule 3567
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \sqrt {e \sec (c+d x)}}{d}+a \int \sqrt {e \sec (c+d x)} \, dx \\ & = \frac {2 i a \sqrt {e \sec (c+d x)}}{d}+\left (a \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 i a \sqrt {e \sec (c+d x)}}{d}+\frac {2 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {2 a \left (i+\sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right ) \sqrt {e \sec (c+d x)}}{d} \]
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Time = 7.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.60
method | result | size |
parts | \(-\frac {2 i a \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{d}+\frac {2 i a \sqrt {e \sec \left (d x +c \right )}}{d}\) | \(96\) |
default | \(-\frac {2 i a \left (F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-1\right ) \sqrt {e \sec \left (d x +c \right )}}{d}\) | \(132\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=-\frac {2 \, {\left (-i \, \sqrt {2} a \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + i \, \sqrt {2} a \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{d} \]
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\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=i a \left (\int \left (- i \sqrt {e \sec {\left (c + d x \right )}}\right )\, dx + \int \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx\right ) \]
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\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\int { \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
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Exception generated. \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \]
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Time = 4.52 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx=\frac {2\,a\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+1{}\mathrm {i}\right )\,\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}}{d} \]
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