Integrand size = 31, antiderivative size = 117 \[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (b \cos (c+d x))^{3+n} \sin (c+d x)}{b^3 d (4+n)}-\frac {(C (3+n)+A (4+n)) (b \cos (c+d x))^{3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) (4+n) \sqrt {\sin ^2(c+d x)}} \]
C*(b*cos(d*x+c))^(3+n)*sin(d*x+c)/b^3/d/(4+n)-(C*(3+n)+A*(4+n))*(b*cos(d*x +c))^(3+n)*hypergeom([1/2, 3/2+1/2*n],[5/2+1/2*n],cos(d*x+c)^2)*sin(d*x+c) /b^3/d/(3+n)/(4+n)/(sin(d*x+c)^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04 \[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {\cos ^2(c+d x) (b \cos (c+d x))^n \cot (c+d x) \left (A (5+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right )+C (3+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+n}{2},\frac {7+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (3+n) (5+n)} \]
-((Cos[c + d*x]^2*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(5 + n)*Hypergeometri c2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] + C*(3 + n)*Cos[c + d*x]^2 *Hypergeometric2F1[1/2, (5 + n)/2, (7 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + n)*(5 + n)))
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2030, 3042, 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.
\(\displaystyle \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (b \cos (c+d x))^{n+2} \left (C \cos ^2(c+d x)+A\right )dx}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n+2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{b^2}\) |
\(\Big \downarrow \) 3493 |
\(\displaystyle \frac {\left (A+\frac {C (n+3)}{n+4}\right ) \int (b \cos (c+d x))^{n+2}dx+\frac {C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b d (n+4)}}{b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (A+\frac {C (n+3)}{n+4}\right ) \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n+2}dx+\frac {C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b d (n+4)}}{b^2}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {\frac {C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b d (n+4)}-\frac {\left (A+\frac {C (n+3)}{n+4}\right ) \sin (c+d x) (b \cos (c+d x))^{n+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2},\frac {n+5}{2},\cos ^2(c+d x)\right )}{b d (n+3) \sqrt {\sin ^2(c+d x)}}}{b^2}\) |
((C*(b*Cos[c + d*x])^(3 + n)*Sin[c + d*x])/(b*d*(4 + n)) - ((A + (C*(3 + n ))/(4 + n))*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(3 + n)*Sqrt[Sin[c + d*x]^2]))/ b^2
3.2.83.3.1 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]
\[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
\[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]