Integrand size = 26, antiderivative size = 86 \[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=-\frac {i 2^{-2-\frac {m}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {6+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a^2 d m} \]
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Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3586, 3604, 72, 71} \[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=-\frac {i 2^{-\frac {m}{2}-2} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {m+6}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{a^2 d m} \]
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Rule 71
Rule 72
Rule 3586
Rule 3596
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \left ((e \cos (c+d x))^m (e \sec (c+d x))^m\right ) \int \frac {(e \sec (c+d x))^{-m}}{(a+i a \tan (c+d x))^2} \, dx \\ & = \left ((e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \int (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-2-\frac {m}{2}} \, dx \\ & = \frac {\left (a^2 (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{m/2}\right ) \text {Subst}\left (\int (a-i a x)^{-1-\frac {m}{2}} (a+i a x)^{-3-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (2^{-3-\frac {m}{2}} (e \cos (c+d x))^m (a-i a \tan (c+d x))^{m/2} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{m/2}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-3-\frac {m}{2}} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {i 2^{-2-\frac {m}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {6+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{a^2 d m} \\ \end{align*}
Time = 3.56 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.16 \[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=-\frac {i 2^{-m/2} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-2-\frac {m}{2},\frac {2+m}{2},-1-\frac {m}{2},\frac {1}{2} (1+i \tan (c+d x))\right ) (1-i \tan (c+d x))^{m/2}}{a^2 d (4+m) (-i+\tan (c+d x))^2} \]
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\[\int \frac {\left (e \cos \left (d x +c \right )\right )^{m}}{\left (a +i a \tan \left (d x +c \right )\right )^{2}}d x\]
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\[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {\left (e \cos {\left (c + d x \right )}\right )^{m}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{m}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^m}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^m}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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