Integrand size = 28, antiderivative size = 150 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3854, 3856, 2719} \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {4 i e^2}{13 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{9/2}}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}} \]
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Rule 2719
Rule 3581
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (9 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{9/2}} \, dx}{13 a^2} \\ & = \frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {7 \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{13 a^2} \\ & = \frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {21 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{65 a^2 e^2} \\ & = \frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {21 \int \sqrt {\cos (c+d x)} \, dx}{65 a^2 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \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 = 2.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {(\cos (2 (c+d x))-i \sin (2 (c+d x))) \left (88 i+416 i \cos (2 (c+d x))-8 i \cos (4 (c+d x))-\frac {224 i e^{4 i (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}-356 \sin (2 (c+d x))+18 \sin (4 (c+d x))\right )}{520 a^2 d e^2 \sqrt {e \sec (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (156 ) = 312\).
Time = 11.16 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.41
method | result | size |
default | \(-\frac {2 i \left (10 i \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )+10 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 \left (\cos ^{7}\left (d x +c \right )\right )+5 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-10 \left (\cos ^{6}\left (d x +c \right )\right )+5 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-21 \cos \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}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+21 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 )+7 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-42 \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}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+42 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 \sin \left (d x +c \right )-21 \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}}+21 \sec \left (d x +c \right ) \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}}\right )}{65 a^{2} d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, e^{2}}\) | \(511\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-13 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 373 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 474 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 118 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 35 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 672 i \, \sqrt {2} \sqrt {e} e^{\left (7 i \, d x + 7 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{1040 \, a^{2} d e^{3}} \]
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\[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (c + d x \right )} - 2 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan {\left (c + d x \right )} - \left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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