\(\int (b \cos (c+d x))^n (A+C \cos ^2(c+d x)) \, dx\) [185]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}-\frac {(C (1+n)+A (2+n)) (b \cos (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) (2+n) \sqrt {\sin ^2(c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(b \cos (c+d x))^n \cot (c+d x) \left (A (3+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right )+C (1+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+n) (3+n)} \] Input:

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

Output:

-(((b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(3 + n)*Hypergeometric2F1[1/2, (1 + 
n)/2, (3 + n)/2, Cos[c + d*x]^2] + C*(1 + n)*Cos[c + d*x]^2*Hypergeometric 
2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*( 
1 + n)*(3 + n)))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3493

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

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

Input:

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

Output:

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

Defintions of rubi rules used

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

b**n*(int(cos(c + d*x)**n,x)*a + int(cos(c + d*x)**n*cos(c + d*x)**2,x)*c)