Integrand size = 28, antiderivative size = 126 \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\frac {10 \cos ^{\frac {9}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}}+\frac {10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac {4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.28 (sec) , antiderivative size = 126, 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 = {3596, 3581, 3853, 3856, 2720} \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\frac {10 \cos ^{\frac {9}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}}+\frac {10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac {4 i \cos ^2(c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{9/2}} \]
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Rule 2720
Rule 3581
Rule 3596
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^2} \, dx}{(e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}} \\ & = -\frac {4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^2\right ) \int (e \sec (c+d x))^{5/2} \, dx}{a^2 (e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}} \\ & = \frac {10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac {4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \sqrt {e \sec (c+d x)} \, dx}{3 a^2 (e \cos (c+d x))^{9/2} (e \sec (c+d x))^{9/2}} \\ & = \frac {10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac {4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 \cos ^{\frac {9}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a^2 (e \cos (c+d x))^{9/2}} \\ & = \frac {10 \cos ^{\frac {9}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{9/2}}+\frac {10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (e \cos (c+d x))^{9/2}}-\frac {4 i \cos ^2(c+d x)}{d (e \cos (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \left (-6 i \cos (c+d x)+5 \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\sin (c+d x)\right )}{3 a^2 d e^3 (e \cos (c+d x))^{3/2}} \]
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Time = 4.56 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {-\frac {20 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-8 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {10 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}}{3}+4 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{4} d}\) | \(208\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (5 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 7 i \, e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 5 \, {\left (i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, {\left (a^{2} d e^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} d e^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{5}\right )}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{9/2} (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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