Integrand size = 28, antiderivative size = 115 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {6 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3853, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {6 e^3 \sin (c+d x) \sqrt {e \sec (c+d x)}}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rule 2719
Rule 3581
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {\left (3 e^2\right ) \int (e \sec (c+d x))^{3/2} \, dx}{a^2} \\ & = -\frac {6 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{a^2} \\ & = -\frac {6 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^4\right ) \int \sqrt {\cos (c+d x)} \, dx}{a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {6 e^3 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{d \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {2 i e^3 e^{-i (c+d x)} \left (-1+3 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )\right ) \sqrt {e \sec (c+d x)}}{a^2 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (131 ) = 262\).
Time = 8.13 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.75
method | result | size |
default | \(-\frac {2 \left (3 i \left (\cos ^{2}\left (d x +c \right )\right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i \left (\cos ^{2}\left (d x +c \right )\right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+6 i \cos \left (d x +c \right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-6 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+3 i E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-2 i \left (\cos ^{2}\left (d x +c \right )\right )-2 i \cos \left (d x +c \right )-2 \sin \left (d x +c \right ) \cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{3}}{a^{2} d \left (\cos \left (d x +c \right )+1\right )}\) | \(431\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (-3 i \, \sqrt {2} e^{\frac {7}{2}} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-3 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, e^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2} d} \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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