Integrand size = 28, antiderivative size = 152 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {6 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.15 (sec) , antiderivative size = 152, 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, 3853, 3856, 2720} \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {6 e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {6 e^5 \sin (c+d x) (e \sec (c+d x))^{3/2}}{7 a^2 d}+\frac {18 e^3 \sin (c+d x) (e \sec (c+d x))^{7/2}}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rule 2720
Rule 3581
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (9 e^2\right ) \int (e \sec (c+d x))^{9/2} \, dx}{5 a^2} \\ & = \frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (9 e^4\right ) \int (e \sec (c+d x))^{5/2} \, dx}{7 a^2} \\ & = \frac {6 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^6\right ) \int \sqrt {e \sec (c+d x)} \, dx}{7 a^2} \\ & = \frac {6 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^6 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2} \\ & = \frac {6 e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {6 e^5 (e \sec (c+d x))^{3/2} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \sec (c+d x))^{7/2} \sin (c+d x)}{35 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{9/2}}{5 d \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.56 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\frac {e^6 \sec ^3(c+d x) \sqrt {e \sec (c+d x)} \left (-56 i \cos (c+d x)+60 \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-5 \sin (c+d x)+15 \sin (3 (c+d x))\right )}{70 a^2 d} \]
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Time = 8.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {2 i e^{6} \sqrt {e \sec \left (d x +c \right )}\, \left (15 F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )+15 F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+15 i \tan \left (d x +c \right )+14 \left (\sec ^{2}\left (d x +c \right )\right )-5 i \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )\right )}{35 a^{2} d}\) | \(174\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32 \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (15 i \, e^{6} e^{\left (6 i \, d x + 6 i \, c\right )} + 51 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} + 61 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i \, e^{6}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 15 \, \sqrt {2} {\left (i \, e^{6} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, e^{6} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{6} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{6}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{35 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {13}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{13/2}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{13/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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