Integrand size = 31, antiderivative size = 127 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=-\frac {b^3 C (b \cos (c+d x))^{-3+n} \sin (c+d x)}{d (2-n)}+\frac {b^3 (A (2-n)+C (3-n)) (b \cos (c+d x))^{-3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-3+n),\frac {1}{2} (-1+n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) (3-n) \sqrt {\sin ^2(c+d x)}} \]
-b^3*C*(b*cos(d*x+c))^(-3+n)*sin(d*x+c)/d/(2-n)+b^3*(A*(2-n)+C*(3-n))*(b*c os(d*x+c))^(-3+n)*hypergeom([1/2, -3/2+1/2*n],[-1/2+1/2*n],cos(d*x+c)^2)*s in(d*x+c)/d/(n^2-5*n+6)/(sin(d*x+c)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=-\frac {(b \cos (c+d x))^n \csc (c+d x) \left (A (-1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-3+n),\frac {1}{2} (-1+n),\cos ^2(c+d x)\right )+C (-3+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+n),\frac {1+n}{2},\cos ^2(c+d x)\right )\right ) \sec ^3(c+d x) \sqrt {\sin ^2(c+d x)}}{d (-3+n) (-1+n)} \]
-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2] + C*(-3 + n)*Cos[c + d*x]^2*Hypergeome tric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^3*Sqrt[S in[c + d*x]^2])/(d*(-3 + n)*(-1 + n)))
Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, 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.
\(\displaystyle \int \sec ^4(c+d x) \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right ) \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^n}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle b^4 \int \left (b \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{n-4} \left (C \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2+A\right )dx\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle b^4 \left (\left (A+\frac {C (3-n)}{2-n}\right ) \int (b \cos (c+d x))^{n-4}dx-\frac {C \sin (c+d x) (b \cos (c+d x))^{n-3}}{b d (2-n)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^4 \left (\left (A+\frac {C (3-n)}{2-n}\right ) \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n-4}dx-\frac {C \sin (c+d x) (b \cos (c+d x))^{n-3}}{b d (2-n)}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle b^4 \left (\frac {\left (A+\frac {C (3-n)}{2-n}\right ) \sin (c+d x) (b \cos (c+d x))^{n-3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n-3}{2},\frac {n-1}{2},\cos ^2(c+d x)\right )}{b d (3-n) \sqrt {\sin ^2(c+d x)}}-\frac {C \sin (c+d x) (b \cos (c+d x))^{n-3}}{b d (2-n)}\right )\) |
b^4*(-((C*(b*Cos[c + d*x])^(-3 + n)*Sin[c + d*x])/(b*d*(2 - n))) + ((A + ( C*(3 - n))/(2 - n))*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(3 - n)*Sqrt[Sin[c + d*x]^2]))
3.2.89.3.1 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]
\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{4}\left (d x +c \right )\right )d x\]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4} \,d x } \]
\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^4} \,d x \]