Integrand size = 26, antiderivative size = 86 \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=-\frac {i 2^{2-\frac {m}{2}} a^2 (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+m),-\frac {m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{d m} \]
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Time = 0.25 (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} \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=-\frac {i a^2 2^{2-\frac {m}{2}} (1+i \tan (c+d x))^{m/2} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (\frac {m-2}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]
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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 (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)^{1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (2^{1-\frac {m}{2}} a^3 (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 )^{1-\frac {m}{2}} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {i 2^{2-\frac {m}{2}} a^2 (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+m),-\frac {m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{d m} \\ \end{align*}
Time = 9.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=-\frac {i 2^{2-\frac {m}{2}} a^2 (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+m),-\frac {m}{2},1-\frac {m}{2},-\frac {1}{2} i (i+\tan (c+d x))\right ) (1+i \tan (c+d x))^{m/2}}{d m} \]
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\[\int \left (e \cos \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{2}d x\]
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\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=- a^{2} \left (\int \left (- \left (e \cos {\left (c + d x \right )}\right )^{m}\right )\, dx + \int \left (e \cos {\left (c + d x \right )}\right )^{m} \tan ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \left (e \cos {\left (c + d x \right )}\right )^{m} \tan {\left (c + d x \right )}\right )\, dx\right ) \]
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\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^m (a+i a \tan (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
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