\(\int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx\) [772]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx=-\frac {a \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {b \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt {\sin ^2(c+d x)}} \] Output:

-a*cos(d*x+c)^(1+m)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],cos(d*x+c)^2)*s 
in(d*x+c)/d/(1+m)/(sin(d*x+c)^2)^(1/2)-b*cos(d*x+c)^(2+m)*hypergeom([1/2, 
1+1/2*m],[2+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(2+m)/(sin(d*x+c)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx=-\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \left (a (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right )+b (1+m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+m) (2+m)} \] Input:

Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x]),x]
 

Output:

-((Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(a*(2 + m)*Hypergeometric2F1[1/2, (1 
+ m)/2, (3 + m)/2, Cos[c + d*x]^2] + b*(1 + m)*Cos[c + d*x]*Hypergeometric 
2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*( 
1 + m)*(2 + m)))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {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.

\(\displaystyle \int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 3227

\(\displaystyle a \int \cos ^m(c+d x)dx+b \int \cos ^{m+1}(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^mdx+b \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+1}dx\)

\(\Big \downarrow \) 3122

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

Input:

Int[Cos[c + d*x]^m*(a + b*Cos[c + d*x]),x]
 

Output:

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

Maple [F]

\[\int \cos \left (d x +c \right )^{m} \left (a +\cos \left (d x +c \right ) b \right )d x\]

Input:

int(cos(d*x+c)^m*(a+cos(d*x+c)*b),x)
 

Output:

int(cos(d*x+c)^m*(a+cos(d*x+c)*b),x)
 

Fricas [F]

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

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c)),x, algorithm="fricas")
 

Output:

integral((b*cos(d*x + c) + a)*cos(d*x + c)^m, x)
 

Sympy [F]

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

integrate(cos(d*x+c)**m*(a+b*cos(d*x+c)),x)
 

Output:

Integral((a + b*cos(c + d*x))*cos(c + d*x)**m, x)
 

Maxima [F]

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

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c)),x, algorithm="maxima")
 

Output:

integrate((b*cos(d*x + c) + a)*cos(d*x + c)^m, x)
 

Giac [F]

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

integrate(cos(d*x+c)^m*(a+b*cos(d*x+c)),x, algorithm="giac")
 

Output:

integrate((b*cos(d*x + c) + a)*cos(d*x + c)^m, x)
 

Mupad [F(-1)]

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

int(cos(c + d*x)^m*(a + b*cos(c + d*x)),x)
 

Output:

int(cos(c + d*x)^m*(a + b*cos(c + d*x)), x)
 

Reduce [F]

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

int(cos(d*x+c)^m*(a+b*cos(d*x+c)),x)
 

Output:

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