\(\int \cos (c+d x) (b \cos (c+d x))^n (A+C \cos ^2(c+d x)) \, dx\) [184]

Optimal result

Integrand size = 29, antiderivative size = 117 \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}-\frac {(C (2+n)+A (3+n)) (b \cos (c+d x))^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) (3+n) \sqrt {\sin ^2(c+d x)}} \] Output:

C*(b*cos(d*x+c))^(2+n)*sin(d*x+c)/b^2/d/(3+n)-(C*(2+n)+A*(3+n))*(b*cos(d*x 
+c))^(2+n)*hypergeom([1/2, 1+1/2*n],[2+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/b^2 
/d/(2+n)/(3+n)/(sin(d*x+c)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {\cos (c+d x) (b \cos (c+d x))^n \cot (c+d x) \left (A (4+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right )+C (2+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+n}{2},\frac {6+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (2+n) (4+n)} \] Input:

Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]
 

Output:

-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(4 + n)*Hypergeometric2 
F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] + C*(2 + n)*Cos[c + d*x]^2*H 
ypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + 
d*x]^2])/(d*(2 + n)*(4 + n)))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.

\(\displaystyle \int \cos (c+d x) \left (A+C \cos ^2(c+d x)\right ) (b \cos (c+d x))^n \, dx\)

\(\Big \downarrow \) 2030

\(\displaystyle \frac {\int (b \cos (c+d x))^{n+1} \left (C \cos ^2(c+d x)+A\right )dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n+1} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{b}\)

\(\Big \downarrow \) 3493

\(\displaystyle \frac {\left (A+\frac {C (n+2)}{n+3}\right ) \int (b \cos (c+d x))^{n+1}dx+\frac {C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b d (n+3)}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (A+\frac {C (n+2)}{n+3}\right ) \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n+1}dx+\frac {C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b d (n+3)}}{b}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\frac {C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b d (n+3)}-\frac {\left (A+\frac {C (n+2)}{n+3}\right ) \sin (c+d x) (b \cos (c+d x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\cos ^2(c+d x)\right )}{b d (n+2) \sqrt {\sin ^2(c+d x)}}}{b}\)

Input:

Int[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]
 

Output:

((C*(b*Cos[c + d*x])^(2 + n)*Sin[c + d*x])/(b*d*(3 + n)) - ((A + (C*(2 + n 
))/(3 + n))*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 
+ n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + n)*Sqrt[Sin[c + d*x]^2]))/ 
b
 

Defintions of rubi rules used

rule 2030

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]
 

rule 3493

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]
 

Maple [F]

\[\int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \cos \left (d x +c \right )^{2}\right )d x\]

Input:

int(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x)
 

Output:

int(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x)
 

Fricas [F]

\[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right ) \,d x } \] Input:

integrate(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x, algorithm="fri 
cas")
 

Output:

integral((C*cos(d*x + c)^3 + A*cos(d*x + c))*(b*cos(d*x + c))^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)*(b*cos(d*x+c))**n*(A+C*cos(d*x+c)**2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right ) \,d x } \] Input:

integrate(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x, algorithm="max 
ima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c))^n*cos(d*x + c), x)
 

Giac [F]

\[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right ) \,d x } \] Input:

integrate(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x, algorithm="gia 
c")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c))^n*cos(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \] Input:

int(cos(c + d*x)*(A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^n,x)
 

Output:

int(cos(c + d*x)*(A + C*cos(c + d*x)^2)*(b*cos(c + d*x))^n, x)
 

Reduce [F]

\[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=b^{n} \left (\left (\int \cos \left (d x +c \right )^{n} \cos \left (d x +c \right )d x \right ) a +\left (\int \cos \left (d x +c \right )^{n} \cos \left (d x +c \right )^{3}d x \right ) c \right ) \] Input:

int(cos(d*x+c)*(b*cos(d*x+c))^n*(A+C*cos(d*x+c)^2),x)
 

Output:

b**n*(int(cos(c + d*x)**n*cos(c + d*x),x)*a + int(cos(c + d*x)**n*cos(c + 
d*x)**3,x)*c)