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

Optimal. Leaf size=55 \[ \frac{2 \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]

[Out]

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

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Rubi [A]  time = 0.0384962, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2750, 2648} \[ \frac{2 \sin (c+d x)}{3 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2750

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)]

Rule 2648

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]

Rubi steps

\begin{align*} \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 \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]  time = 0.112811, size = 60, normalized size = 1.09 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-3 \sin \left (c+\frac{d x}{2}\right )+2 \sin \left (c+\frac{3 d x}{2}\right )+3 \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right )}{12 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^3*(3*Sin[(d*x)/2] - 3*Sin[c + (d*x)/2] + 2*Sin[c + (3*d*x)/2]))/(12*a^2*d)

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Maple [A]  time = 0.036, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{2\,{a}^{2}d} \left ( -{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17205, size = 63, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.69925, size = 126, normalized size = 2.29 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.07279, size = 48, normalized size = 0.87 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.19589, size = 42, normalized size = 0.76 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)