Integrand size = 28, antiderivative size = 92 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac {4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3596, 3581, 3856, 2720} \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac {4 i \cos ^2(c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{5/2}} \]
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Rule 2720
Rule 3581
Rule 3596
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {e^2 \int \sqrt {e \sec (c+d x)} \, dx}{3 a^2 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}} \\ & = \frac {4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {\cos ^{\frac {5}{2}}(c+d x) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a^2 (e \cos (c+d x))^{5/2}} \\ & = -\frac {2 \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a^2 d (e \cos (c+d x))^{5/2}}+\frac {4 i \cos ^2(c+d x)}{3 d (e \cos (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} (\cos (d x)+i \sin (d x))^2 \left (\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 c)+i \sin (2 c))+2 \sqrt {\cos (c+d x)} (-i \cos (c-d x)+\sin (c-d x))\right )}{3 a^2 d (e \cos (c+d x))^{5/2} (-i+\tan (c+d x))^2} \]
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Time = 3.52 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86
method | result | size |
default | \(-\frac {2 \left (-8 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}-2 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 e^{2} a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(171\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=-\frac {2 \, {\left (-i \, \sqrt {2} \sqrt {e} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) - 2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{3 \, a^{2} d e^{3}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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