\(\int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx\) [402]

Optimal result

Integrand size = 21, antiderivative size = 229 \[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {2 (1-m) \cos ^{1+m}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {\cos ^{1+m}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(1-2 m) m \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{3 a^2 d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {2 (1-m) (1+m) \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{3 a^2 d (2+m) \sqrt {\sin ^2(c+d x)}} \] Output:

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

Mathematica [A] (verified)

Time = 3.15 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \left (-\sin ^2(c+d x)-\frac {(1+\cos (c+d x)) \left (-2 (-1+m) (1+m) (2+m) \sin ^2(c+d x)-(1+\cos (c+d x)) \left ((1-2 m) m (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right )+2 (-1+m) (1+m)^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}\right )}{(1+m) (2+m)}\right )}{3 a^2 d (1+\cos (c+d x))^2} \] Input:

Integrate[Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3245, 3042, 3457, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

\(\displaystyle \frac {\int \frac {\cos ^m(c+d x) (a (2-m)+a m \cos (c+d x))}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^m \left (a (2-m)+a m \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {\int -\cos ^m(c+d x) \left (a^2 (1-2 m) m-2 a^2 (1-m) (m+1) \cos (c+d x)\right )dx}{a^2}-\frac {2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \cos ^m(c+d x) \left (a^2 (1-2 m) m-2 a^2 (1-m) (m+1) \cos (c+d x)\right )dx}{a^2}-\frac {2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^m \left (a^2 (1-2 m) m-2 a^2 (1-m) (m+1) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}-\frac {2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {-\frac {a^2 (1-2 m) m \int \cos ^m(c+d x)dx-2 a^2 (1-m) (m+1) \int \cos ^{m+1}(c+d x)dx}{a^2}-\frac {2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {a^2 (1-2 m) m \int \sin \left (c+d x+\frac {\pi }{2}\right )^mdx-2 a^2 (1-m) (m+1) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{m+1}dx}{a^2}-\frac {2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (\cos (c+d x)+1)}}{3 a^2}-\frac {\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}\)

\(\Big \downarrow \) 3122

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

Input:

Int[Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3245

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ 
m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( 
a*(2*m + 1)*(b*c - a*d))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + 
f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] 
, 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] && LtQ[m, -1] &&  !GtQ[n, 0] && (Intege 
rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
 

rule 3457

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

Maple [F]

\[\int \frac {\cos \left (d x +c \right )^{m}}{\left (a +a \cos \left (d x +c \right )\right )^{2}}d x\]

Input:

int(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x)
 

Output:

int(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x)
 

Fricas [F]

\[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x, algorithm="fricas")
 

Output:

integral(cos(d*x + c)^m/(a^2*cos(d*x + c)^2 + 2*a^2*cos(d*x + c) + a^2), x 
)
 

Sympy [F]

\[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\int \frac {\cos ^{m}{\left (c + d x \right )}}{\cos ^{2}{\left (c + d x \right )} + 2 \cos {\left (c + d x \right )} + 1}\, dx}{a^{2}} \] Input:

integrate(cos(d*x+c)**m/(a+a*cos(d*x+c))**2,x)
 

Output:

Integral(cos(c + d*x)**m/(cos(c + d*x)**2 + 2*cos(c + d*x) + 1), x)/a**2
 

Maxima [F]

\[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x, algorithm="maxima")
 

Output:

integrate(cos(d*x + c)^m/(a*cos(d*x + c) + a)^2, x)
 

Giac [F]

\[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate(cos(d*x + c)^m/(a*cos(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^m}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \] Input:

int(cos(c + d*x)^m/(a + a*cos(c + d*x))^2,x)
 

Output:

int(cos(c + d*x)^m/(a + a*cos(c + d*x))^2, x)
 

Reduce [F]

\[ \int \frac {\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\int \frac {\cos \left (d x +c \right )^{m}}{\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1}d x}{a^{2}} \] Input:

int(cos(d*x+c)^m/(a+a*cos(d*x+c))^2,x)
                                                                                    
                                                                                    
 

Output:

int(cos(c + d*x)**m/(cos(c + d*x)**2 + 2*cos(c + d*x) + 1),x)/a**2