Integrand size = 23, antiderivative size = 266 \[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\frac {d \operatorname {AppellF1}\left (1-n,\frac {1-n}{2},\frac {1-n}{2},2-n,\frac {a+b}{a+b \sec (e+f x)},\frac {a-b}{a+b \sec (e+f x)}\right ) \left (-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}\right )^{\frac {1-n}{2}} \left (\frac {b (1+\sec (e+f x))}{a+b \sec (e+f x)}\right )^{\frac {1-n}{2}} (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}+\frac {1}{2} (-1+n)}}{a f (1-n)}-\frac {d \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}+\frac {1+n}{2}}}{a f (1+n)} \]
d*AppellF1(1-n,1/2-1/2*n,1/2-1/2*n,2-n,(a-b)/(a+b*sec(f*x+e)),(a+b)/(a+b*s ec(f*x+e)))*(-b*(1-sec(f*x+e))/(a+b*sec(f*x+e)))^(1/2-1/2*n)*(b*(1+sec(f*x +e))/(a+b*sec(f*x+e)))^(1/2-1/2*n)*(d*tan(f*x+e))^(-1+n)/a/f/(1-n)+d*hyper geom([1, 1/2+1/2*n],[3/2+1/2*n],-tan(f*x+e)^2)*(d*tan(f*x+e))^(-1+n)*tan(f *x+e)^2/a/f/(1+n)
Leaf count is larger than twice the leaf count of optimal. \(786\) vs. \(2(266)=532\).
Time = 4.22 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.95 \[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\frac {2 \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right ) (d \tan (e+f x))^n}{f (a+b \sec (e+f x)) \left (\left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )-16 n \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x) \sec (e+f x) \sin ^5\left (\frac {1}{2} (e+f x)\right )+2 n \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \csc (e+f x) \sec (e+f x) \tan \left (\frac {1}{2} (e+f x)\right )-\frac {2 (1+n) \left ((a-b) b \operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+(a+b)^2 \left (\operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+b (a+b) n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{(a+b) (3+n)}\right )} \]
(2*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[ (e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^ 2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]*(d*Tan[e + f*x] )^n)/(f*(a + b*Sec[e + f*x])*(((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2 , Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, ( 3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Sec[( e + f*x)/2]^2 - 16*n*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Cos[(e + f*x)/ 2]*Csc[e + f*x]^3*Sec[e + f*x]*Sin[(e + f*x)/2]^5 + 2*n*((a + b)*AppellF1[ (1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*A ppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Csc[e + f*x]*Sec[e + f*x]*Tan[(e + f*x)/2] - (2*(1 + n)*((a - b)*b*AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + (a + b)^2*(AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + n)/2 , 1 + n, 1, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]) + b*(a + b)*n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(e + f*x)/2]^2, ((a - b) *Tan[(e + f*x)/2]^2)/(a + b)])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2)/((a + b)*(3 + n))))
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.
\(\displaystyle \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (-d \cot \left (e+f x+\frac {\pi }{2}\right )\right )^n}{a+b \csc \left (e+f x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4387 |
\(\displaystyle \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)}dx\) |
3.4.47.3.1 Defintions of rubi rules used
Int[((a_.) + csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_.)*(cot[(c_.) + (d_.)*(x_)]* (e_.))^(m_.), x_Symbol] :> Unintegrable[(e*Cot[c + d*x])^m*(a + b*Csc[c + d *x])^n, x] /; FreeQ[{a, b, c, d, e, m, n}, x]
\[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{a +b \sec \left (f x +e \right )}d x\]
\[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{a + b \sec {\left (e + f x \right )}}\, dx \]
\[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a} \,d x } \]
\[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{b+a\,\cos \left (e+f\,x\right )} \,d x \]