Integrand size = 26, antiderivative size = 124 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d} \]
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Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567, 3854, 3856, 2720} \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=-\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {10 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (e \cos (c+d x))^{7/2}}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a \tan (c+d x) (e \cos (c+d x))^{7/2}}{7 d}+\frac {10 a \tan (c+d x) \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{21 d} \]
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Rule 2720
Rule 3567
Rule 3596
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{7/2}} \, dx \\ & = -\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\left (a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx \\ & = -\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {\left (5 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2} \\ & = -\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d}+\frac {\left (5 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \sqrt {e \sec (c+d x)} \, dx}{21 e^4} \\ & = -\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d}+\frac {\left (5 a (e \cos (c+d x))^{7/2}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \cos ^{\frac {7}{2}}(c+d x)} \\ & = -\frac {2 i a (e \cos (c+d x))^{7/2}}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (e \cos (c+d x))^{7/2} \tan (c+d x)}{7 d}+\frac {10 a (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \tan (c+d x)}{21 d} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\frac {a e^3 \sqrt {e \cos (c+d x)} (\cos (d x)-i \sin (d x)) \left (10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (c+d x)-i \sin (c+d x))+\sqrt {\cos (c+d x)} (-8 i+2 i \cos (2 (c+d x))+5 \sin (2 (c+d x)))\right ) (\cos (c+2 d x)+i \sin (c+2 d x))}{21 d \sqrt {\cos (c+d x)}} \]
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Time = 9.94 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85
method | result | size |
parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 i a \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d}\) | \(229\) |
default | \(-\frac {2 a \,e^{4} \left (48 i \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+72 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}+3 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(241\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\frac {{\left (-20 i \, \sqrt {2} a e^{\frac {7}{2}} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {\frac {1}{2}} {\left (-3 i \, a e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 16 i \, a e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{42 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
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\[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
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