\(\int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx\) [47]

Optimal result

Integrand size = 19, antiderivative size = 29 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {x}{a}-\frac {\sin (c+d x)}{d (a+a \cos (c+d x))} \]

[Out]

x/a-sin(d*x+c)/d/(a+a*cos(d*x+c))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2814, 2727} \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)} \]

[In]

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

[Out]

x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))

Rule 2727

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

Rule 2814

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(29)=58\).

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\sin (c+d x) \left (\arcsin (\cos (c+d x)) (1+\cos (c+d x))+\sqrt {\sin ^2(c+d x)}\right )}{a d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {d x -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}\)	\(23\)
risch	\(\frac {x}{a}-\frac {2 i}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\)	\(29\)
derivativedivides	\(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(32\)
default	\(\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(32\)
norman	\(\frac {\frac {x}{a}+\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\)	\(75\)

[In]

int(cos(d*x+c)/(a+cos(d*x+c)*a),x,method=_RETURNVERBOSE)

[Out]

(d*x-tan(1/2*d*x+1/2*c))/a/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {d x \cos \left (d x + c\right ) + d x - \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]

[In]

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

[Out]

(d*x*cos(d*x + c) + d*x - sin(d*x + c))/(a*d*cos(d*x + c) + a*d)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\begin {cases} \frac {x}{a} - \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{a \cos {\left (c \right )} + a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/a - tan(c/2 + d*x/2)/(a*d), Ne(d, 0)), (x*cos(c)/(a*cos(c) + a), True))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]

[In]

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

[Out]

(2*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - sin(d*x + c)/(a*(cos(d*x + c) + 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\frac {d x + c}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a}}{d} \]

[In]

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

[Out]

((d*x + c)/a - tan(1/2*d*x + 1/2*c)/a)/d

Mupad [B] (verification not implemented)

Time = 14.64 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx=\frac {x}{a}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]

[In]

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

[Out]

x/a - tan(c/2 + (d*x)/2)/(a*d)