Integrand size = 24, antiderivative size = 96 \[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=-\frac {\operatorname {AppellF1}\left (n,\frac {1}{2}-m,\frac {1}{2},1+n,\sec (e+f x),-\sec (e+f x)\right ) (1-\sec (e+f x))^{-\frac {1}{2}-m} (d \sec (e+f x))^n (a-a \sec (e+f x))^m \tan (e+f x)}{f n \sqrt {1+\sec (e+f x)}} \]
-AppellF1(n,1/2,1/2-m,1+n,-sec(f*x+e),sec(f*x+e))*(1-sec(f*x+e))^(-1/2-m)* (d*sec(f*x+e))^n*(a-a*sec(f*x+e))^m*tan(f*x+e)/f/n/(1+sec(f*x+e))^(1/2)
\[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx \]
Time = 0.36 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 4315, 3042, 4314, 150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a-a \sec (e+f x))^m (d \sec (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a-a \csc \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^ndx\) |
\(\Big \downarrow \) 4315 |
\(\displaystyle (1-\sec (e+f x))^{-m} (a-a \sec (e+f x))^m \int (1-\sec (e+f x))^m (d \sec (e+f x))^ndx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (1-\sec (e+f x))^{-m} (a-a \sec (e+f x))^m \int \left (1-\csc \left (e+f x+\frac {\pi }{2}\right )\right )^m \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^ndx\) |
\(\Big \downarrow \) 4314 |
\(\displaystyle -\frac {d \tan (e+f x) (1-\sec (e+f x))^{-m-\frac {1}{2}} (a-a \sec (e+f x))^m \int \frac {(1-\sec (e+f x))^{m-\frac {1}{2}} (d \sec (e+f x))^{n-1}}{\sqrt {\sec (e+f x)+1}}d\sec (e+f x)}{f \sqrt {\sec (e+f x)+1}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle -\frac {\tan (e+f x) (1-\sec (e+f x))^{-m-\frac {1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n \operatorname {AppellF1}\left (n,\frac {1}{2}-m,\frac {1}{2},n+1,\sec (e+f x),-\sec (e+f x)\right )}{f n \sqrt {\sec (e+f x)+1}}\) |
-((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(1 - Sec[ e + f*x])^(-1/2 - m)*(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m*Tan[e + f*x ])/(f*n*Sqrt[1 + Sec[e + f*x]]))
3.4.40.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*d*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x ]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(d*x)^(n - 1)*((a + b*x)^(m - 1/2 )/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Csc[e + f*x])^FracPart[m ]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Csc[e + f*x])^m*(d *Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
\[\int \left (d \sec \left (f x +e \right )\right )^{n} \left (a -a \sec \left (f x +e \right )\right )^{m}d x\]
\[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{n} \left (- a \left (\sec {\left (e + f x \right )} - 1\right )\right )^{m}\, dx \]
\[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int { {\left (-a \sec \left (f x + e\right ) + a\right )}^{m} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]
Timed out. \[ \int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx=\int {\left (a-\frac {a}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]