Integrand size = 28, antiderivative size = 105 \[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {i 2^{\frac {1}{2}-\frac {m}{2}} a (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {1+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1+m}{2}}}{d m \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.39 (sec) , antiderivative size = 105, 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, 3586, 3604, 72, 71} \[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {i a 2^{\frac {1}{2}-\frac {m}{2}} (1+i \tan (c+d x))^{\frac {m+1}{2}} (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {m+1}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m \sqrt {a+i a \tan (c+d x)}} \]
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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} \sqrt {a+i a \tan (c+d x)} \, 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))^{\frac {1}{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)^{-\frac {1}{2}-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (2^{-\frac {1}{2}-\frac {m}{2}} a^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 )^{\frac {1}{2}+\frac {m}{2}}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {1}{2}-\frac {m}{2}} (a-i a x)^{-1-\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d \sqrt {a+i a \tan (c+d x)}} \\ & = -\frac {i 2^{\frac {1}{2}-\frac {m}{2}} a (e \cos (c+d x))^m \operatorname {Hypergeometric2F1}\left (-\frac {m}{2},\frac {1+m}{2},1-\frac {m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1+m}{2}}}{d m \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 2.50 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\frac {i 2^{-m} \left (1+e^{2 i (c+d x)}\right )^{\frac {1}{2}-m} \left (e e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-m,\frac {1-m}{2},\frac {3-m}{2},-e^{2 i (c+d x)}\right ) \sqrt {a+i a \tan (c+d x)}}{d (-1+m)} \]
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\[\int \left (e \cos \left (d x +c \right )\right )^{m} \sqrt {a +i a \tan \left (d x +c \right )}d x\]
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\[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\int \left (e \cos {\left (c + d x \right )}\right )^{m} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}\, dx \]
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\[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \left (e \cos \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^m \sqrt {a+i a \tan (c+d x)} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^m\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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