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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2848, 2827, 2722} \[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {m \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(c+d x)\right )}{a d (m+1) \sqrt {\sin ^2(c+d x)}}-\frac {\sin (c+d x) \cos ^m(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {m+2}{2},\cos ^2(c+d x)\right )}{a d \sqrt {\sin ^2(c+d x)}}+\frac {\sin (c+d x) \cos ^m(c+d x)}{d (a \cos (c+d x)+a)} \]

[In]

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

[Out]

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

Rule 2722

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]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 2827

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

Rule 2848

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] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^m(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {m \int \cos ^{-1+m}(c+d x) (a-a \cos (c+d x)) \, dx}{a^2} \\ & = \frac {\cos ^m(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {m \int \cos ^{-1+m}(c+d x) \, dx}{a}-\frac {m \int \cos ^m(c+d x) \, dx}{a} \\ & = \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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\cos ^m(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \left (-((1+m) (-1+\cos (c+d x)))-(1+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m}{2},\frac {2+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+m \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{a d (1+m) (1+\cos (c+d x))} \]

[In]

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

[Out]

(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]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,1,0]%%%} / %%%{2,[0,0,1]%%%} Error: Bad Argument Va
lue

Mupad [F(-1)]

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

[In]

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

[Out]

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