Integrand size = 29, antiderivative size = 100 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {C (b \cos (c+d x))^n \sin (c+d x)}{d (1+n)}-\frac {(A+A n+C n) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d n (1+n) \sqrt {\sin ^2(c+d x)}} \]
C*(b*cos(d*x+c))^n*sin(d*x+c)/d/(1+n)-(A*n+C*n+A)*(b*cos(d*x+c))^n*hyperge om([1/2, 1/2*n],[1+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/d/n/(1+n)/(sin(d*x+c)^2 )^(1/2)
Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {b (b \cos (c+d x))^{-1+n} \cot (c+d x) \left (A (2+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right )+C n \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d n (2+n)} \]
-((b*(b*Cos[c + d*x])^(-1 + n)*Cot[c + d*x]*(A*(2 + n)*Hypergeometric2F1[1 /2, n/2, (2 + n)/2, Cos[c + d*x]^2] + C*n*Cos[c + d*x]^2*Hypergeometric2F1 [1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*n*(2 + n)))
Time = 0.37 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 2030, 3493, 3042, 3122}
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.
\(\displaystyle \int \sec (c+d x) \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right ) \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^n}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle b \int \left (b \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{n-1} \left (C \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2+A\right )dx\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle b \left (\frac {(A n+A+C n) \int (b \cos (c+d x))^{n-1}dx}{n+1}+\frac {C \sin (c+d x) (b \cos (c+d x))^n}{b d (n+1)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \left (\frac {(A n+A+C n) \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n-1}dx}{n+1}+\frac {C \sin (c+d x) (b \cos (c+d x))^n}{b d (n+1)}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle b \left (\frac {C \sin (c+d x) (b \cos (c+d x))^n}{b d (n+1)}-\frac {(A n+A+C n) \sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {n+2}{2},\cos ^2(c+d x)\right )}{b d n (n+1) \sqrt {\sin ^2(c+d x)}}\right )\) |
b*((C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(b*d*(1 + n)) - ((A + A*n + C*n)*(b *Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Si n[c + d*x])/(b*d*n*(1 + n)*Sqrt[Sin[c + d*x]^2]))
3.2.86.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f *(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2) Int[(b*Sin[e + f*x] )^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && !LtQ[m, -1]
\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \sec \left (d x +c \right )d x\]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
Timed out. \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{\cos \left (c+d\,x\right )} \,d x \]