Integrand size = 21, antiderivative size = 106 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {2^{-3+n} \operatorname {AppellF1}\left (-\frac {3}{2},-4+n,1,-\frac {1}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \cot ^3(c+d x) \left (\frac {1}{1+\sec (c+d x)}\right )^{-3+n} (a+a \sec (c+d x))^n}{3 d} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3974} \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {2^{n-3} \cot ^3(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n-3} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (-\frac {3}{2},n-4,1,-\frac {1}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d} \]
[In]
[Out]
Rule 3974
Rubi steps \begin{align*} \text {integral}& = -\frac {2^{-3+n} \operatorname {AppellF1}\left (-\frac {3}{2},-4+n,1,-\frac {1}{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 ) \cot ^3(c+d x) \left (\frac {1}{1+\sec (c+d x)}\right )^{-3+n} (a+a \sec (c+d x))^n}{3 d} \\ \end{align*}
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx \]
[In]
[Out]
\[\int \cot \left (d x +c \right )^{4} \left (a +a \sec \left (d x +c \right )\right )^{n}d x\]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \cot ^{4}{\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
[In]
[Out]