Integrand size = 28, antiderivative size = 132 \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {10 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3581, 3856, 2720} \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {10 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3} \]
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Rule 2720
Rule 3581
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {\left (5 e^2\right ) \int \frac {(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx}{7 a^2} \\ & = \frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \sqrt {e \sec (c+d x)} \, dx}{21 a^4} \\ & = \frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (5 e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^4} \\ & = \frac {10 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\frac {2 e^4 \sec ^4(c+d x) \sqrt {e \sec (c+d x)} \left (5 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i (1+\cos (2 (c+d x))+4 i \sin (2 (c+d x)))\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{21 a^4 d (-i+\tan (c+d x))^4} \]
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Time = 7.63 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {2 e^{4} \left (24 i \left (\cos ^{4}\left (d x +c \right )\right )+5 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+24 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-28 i \left (\cos ^{2}\left (d x +c \right )\right )-16 \sin \left (d x +c \right ) \cos \left (d x +c \right )\right ) \sqrt {e \sec \left (d x +c \right )}}{21 a^{4} d}\) | \(191\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84 \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} e^{\frac {9}{2}} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {2} {\left (5 i \, e^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, e^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{21 \, a^{4} d} \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \,d x \]
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