Integrand size = 25, antiderivative size = 131 \[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\frac {2^{\frac {3}{2}+m} \operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1}{2}+m,1,\frac {3+m}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{\frac {3}{2}+m} \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^{1+m}}{d e (1+m)} \]
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Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3974} \[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\frac {2^{m+\frac {3}{2}} \sqrt {a \sec (c+d x)+a} \left (\frac {1}{\sec (c+d x)+1}\right )^{m+\frac {3}{2}} (e \tan (c+d x))^{m+1} \operatorname {AppellF1}\left (\frac {m+1}{2},m+\frac {1}{2},1,\frac {m+3}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1)} \]
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Rule 3974
Rubi steps \begin{align*} \text {integral}& = \frac {2^{\frac {3}{2}+m} \operatorname {AppellF1}\left (\frac {1+m}{2},\frac {1}{2}+m,1,\frac {3+m}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{\frac {3}{2}+m} \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^{1+m}}{d e (1+m)} \\ \end{align*}
\[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx \]
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\[\int \sqrt {a +a \sec \left (d x +c \right )}\, \left (e \tan \left (d x +c \right )\right )^{m}d x\]
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\[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \left (e \tan {\left (c + d x \right )}\right )^{m}\, dx \]
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\[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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\[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]
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Timed out. \[ \int \sqrt {a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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