3.4.71 \(\int \cos (c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [371]

3.4.71.1 Optimal result

Integrand size = 37, antiderivative size = 187 \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}-\frac {(C (2+n)+A (3+n)) (b \cos (c+d x))^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) (3+n) \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{3+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt {\sin ^2(c+d x)}} \]

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)-B*(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)/(sin(d*x+c)^2 
)^(1/2)
 

3.4.71.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81 \[ \int \cos (c+d x) (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {\cos (c+d x) (b \cos (c+d x))^n \cot (c+d x) \left ((C (2+n)+A (3+n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2},\frac {4+n}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}-(2+n) \left (C \sin ^2(c+d x)-B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )\right )}{d (2+n) (3+n)} \]

input

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

output

-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*((C*(2 + n) + A*(3 + n))*H 
ypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d 
*x]^2] - (2 + n)*(C*Sin[c + d*x]^2 - B*Cos[c + d*x]*Hypergeometric2F1[1/2, 
 (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])))/(d*(2 + n)* 
(3 + n)))
 

3.4.71.3 Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {2030, 3042, 3502, 3042, 3227, 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.

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

\(\Big \downarrow \) 2030

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3227

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\frac {-\frac {(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2},\frac {n+4}{2},\cos ^2(c+d x)\right )}{d (n+2) \sqrt {\sin ^2(c+d x)}}-\frac {B \sin (c+d x) (b \cos (c+d x))^{n+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+3}{2},\frac {n+5}{2},\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)}}}{b (n+3)}+\frac {C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b d (n+3)}}{b}\)

input

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

output

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

3.4.71.3.1 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 3227

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

3.4.71.4 Maple [F]

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

input

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

output

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

3.4.71.5 Fricas [F]

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

input

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

output

integral((C*cos(d*x + c)^3 + B*cos(d*x + c)^2 + A*cos(d*x + c))*(b*cos(d*x 
 + c))^n, x)
 

3.4.71.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.4.71.7 Maxima [F]

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

input

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

output

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

3.4.71.8 Giac [F]

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

input

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

output

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

3.4.71.9 Mupad [F(-1)]

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

input

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

output

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