\(\int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx\) [57]

Optimal result

Integrand size = 19, antiderivative size = 55 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {2 \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )} \]

[Out]

-1/3*sin(d*x+c)/d/(a+a*cos(d*x+c))^2+2/3*sin(d*x+c)/d/(a^2+a^2*cos(d*x+c))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, 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 = {2829, 2727} \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {2 \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]

[In]

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

[Out]

-1/3*Sin[c + d*x]/(d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(3*d*(a^2 + a^2*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 2829

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {2 \int \frac {1}{a+a \cos (c+d x)} \, dx}{3 a} \\ & = -\frac {\sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {2 \sin (c+d x)}{3 d \left (a^2+a^2 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {(1+2 \cos (c+d x)) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{6 a^{2} d}\)	\(31\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}\)	\(32\)
default	\(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}\)	\(32\)
risch	\(\frac {2 i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\)	\(47\)
norman	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{6 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\)	\(76\)

[In]

int(cos(d*x+c)/(a+cos(d*x+c)*a)^2,x,method=_RETURNVERBOSE)

[Out]

-1/6*tan(1/2*d*x+1/2*c)*(tan(1/2*d*x+1/2*c)^2-3)/a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]

[In]

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

[Out]

1/3*(2*cos(d*x + c) + 1)*sin(d*x + c)/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=\begin {cases} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-tan(c/2 + d*x/2)**3/(6*a**2*d) + tan(c/2 + d*x/2)/(2*a**2*d), Ne(d, 0)), (x*cos(c)/(a*cos(c) + a)*
*2, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{6 \, a^{2} d} \]

[In]

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

[Out]

1/6*(3*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^2*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{6 \, a^{2} d} \]

[In]

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

[Out]

-1/6*(tan(1/2*d*x + 1/2*c)^3 - 3*tan(1/2*d*x + 1/2*c))/(a^2*d)

Mupad [B] (verification not implemented)

Time = 14.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.55 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3\right )}{6\,a^2\,d} \]

[In]

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

[Out]

-(tan(c/2 + (d*x)/2)*(tan(c/2 + (d*x)/2)^2 - 3))/(6*a^2*d)