\(\int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx\) [401]

Optimal result

Integrand size = 21, antiderivative size = 156 \[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\cos ^m(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {\cos ^m(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{a d \sqrt {\sin ^2(c+d x)}}+\frac {m \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{a d (1+m) \sqrt {\sin ^2(c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\cos ^m(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \left (-((1+m) (-1+\cos (c+d x)))-(1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+m \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{a d (1+m) (1+\cos (c+d x))} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3248, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3248

\(\displaystyle \frac {m \int \cos ^{m-1}(c+d x) (a-a \cos (c+d x))dx}{a^2}+\frac {\sin (c+d x) \cos ^m(c+d x)}{d (a \cos (c+d x)+a)}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3227

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

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

Input:

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

Output:

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

rule 3248

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/( 
a*f*(a + b*Sin[e + f*x]))), x] + Simp[d*(n/(a*b))   Int[(c + d*Sin[e + f*x] 
)^(n - 1)*(a - b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] & 
& NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (IntegerQ[ 
2*n] || EqQ[c, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {\cos ^{m}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

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

Output:

Integral(cos(c + d*x)**m/(cos(c + d*x) + 1), x)/a
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {\cos \left (d x +c \right )^{m}}{\cos \left (d x +c \right )+1}d x}{a} \] Input:

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

Output:

int(cos(c + d*x)**m/(cos(c + d*x) + 1),x)/a