Integrand size = 25, antiderivative size = 131 \[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2^{-\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1+m}{2},-\frac {3}{2}+m,1,\frac {3+m}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{-\frac {1}{2}+m} (e \tan (c+d x))^{1+m}}{d e (1+m) (a+a \sec (c+d x))^{3/2}} \]
[Out]
Time = 0.12 (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} \[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2^{m-\frac {1}{2}} \left (\frac {1}{\sec (c+d x)+1}\right )^{m-\frac {1}{2}} (e \tan (c+d x))^{m+1} \operatorname {AppellF1}\left (\frac {m+1}{2},m-\frac {3}{2},1,\frac {m+3}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) (a \sec (c+d x)+a)^{3/2}} \]
[In]
[Out]
Rule 3974
Rubi steps \begin{align*} \text {integral}& = \frac {2^{-\frac {1}{2}+m} \operatorname {AppellF1}\left (\frac {1+m}{2},-\frac {3}{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 {1}{2}+m} (e \tan (c+d x))^{1+m}}{d e (1+m) (a+a \sec (c+d x))^{3/2}} \\ \end{align*}
\[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx \]
[In]
[Out]
\[\int \frac {\left (e \tan \left (d x +c \right )\right )^{m}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (e \tan {\left (c + d x \right )}\right )^{m}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]