Integrand size = 28, antiderivative size = 141 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \]
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Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3581, 3582, 3853, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {14 e^5 \sin (c+d x) \sqrt {e \sec (c+d x)}}{a^3 d}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \]
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Rule 2719
Rule 3581
Rule 3582
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}-\frac {\left (7 e^2\right ) \int \frac {(e \sec (c+d x))^{7/2}}{a+i a \tan (c+d x)} \, dx}{a^2} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}-\frac {\left (7 e^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{a^3} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}+\frac {\left (7 e^6\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{a^3} \\ & = \frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2}+\frac {\left (7 e^6\right ) \int \sqrt {\cos (c+d x)} \, dx}{a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 i e^4 (e \sec (c+d x))^{3/2}}{3 a^3 d}-\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{a d (a+i a \tan (c+d x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.85 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.66 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {i e^4 (e \sec (c+d x))^{3/2} \left (35+33 \cos (2 (c+d x))-7 \left (1+e^{2 i (c+d x)}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+9 i \sin (2 (c+d x))\right )}{3 a^3 d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (152 ) = 304\).
Time = 10.35 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.15
method | result | size |
default | \(\frac {2 \sqrt {e \sec \left (d x +c \right )}\, e^{5} \left (21 i \left (\cos ^{2}\left (d x +c \right )\right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-21 i \left (\cos ^{2}\left (d x +c \right )\right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+42 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-42 i \cos \left (d x +c \right ) E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+21 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-21 i E\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+12 i \left (\cos ^{2}\left (d x +c \right )\right )+12 i \cos \left (d x +c \right )+12 \sin \left (d x +c \right ) \cos \left (d x +c \right )+i-9 \sin \left (d x +c \right )+i \sec \left (d x +c \right )\right )}{3 a^{3} d \left (\cos \left (d x +c \right )+1\right )}\) | \(444\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (-21 i \, e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 35 i \, e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, e^{5}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 21 \, \sqrt {2} {\left (-i \, e^{5} e^{\left (3 i \, d x + 3 i \, c\right )} - i \, e^{5} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{3 \, {\left (a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\right )}} \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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