3.179 \(\int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2} \]

[Out]

((I/2)*Cot[c + d*x]^2)/(a^3*d*(I + Cot[c + d*x])^2)

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Rubi [A]  time = 0.0318248, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {3088, 37} \[ \frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

((I/2)*Cot[c + d*x]^2)/(a^3*d*(I + Cot[c + d*x])^2)

Rule 3088

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Dist[d^(-1), Subst[Int[(x^m*(b + a*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]], x] /; FreeQ[
{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] &&  !(GtQ[n, 0] && GtQ[m, 1])

Rule 37

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

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(i a+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{i \cot ^2(c+d x)}{2 a^3 d (i+\cot (c+d x))^2}\\ \end{align*}

Mathematica [B]  time = 0.0593853, size = 77, normalized size = 2.41 \[ \frac{\sin (2 (c+d x))}{4 a^3 d}+\frac{\sin (4 (c+d x))}{8 a^3 d}+\frac{i \cos (2 (c+d x))}{4 a^3 d}+\frac{i \cos (4 (c+d x))}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]

[Out]

((I/4)*Cos[2*(c + d*x)])/(a^3*d) + ((I/8)*Cos[4*(c + d*x)])/(a^3*d) + Sin[2*(c + d*x)]/(4*a^3*d) + Sin[4*(c +
d*x)]/(8*a^3*d)

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Maple [A]  time = 0.115, size = 23, normalized size = 0.7 \begin{align*}{\frac{{\frac{i}{2}}}{d{a}^{3} \left ( i\tan \left ( dx+c \right ) +1 \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.15423, size = 69, normalized size = 2.16 \begin{align*} \frac{i \, \cos \left (4 \, d x + 4 \, c\right ) + 2 i \, \cos \left (2 \, d x + 2 \, c\right ) + \sin \left (4 \, d x + 4 \, c\right ) + 2 \, \sin \left (2 \, d x + 2 \, c\right )}{8 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(I*cos(4*d*x + 4*c) + 2*I*cos(2*d*x + 2*c) + sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))/(a^3*d)

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Fricas [A]  time = 0.462933, size = 86, normalized size = 2.69 \begin{align*} \frac{{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/8*(2*I*e^(2*I*d*x + 2*I*c) + I)*e^(-4*I*d*x - 4*I*c)/(a^3*d)

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Sympy [A]  time = 0.419022, size = 97, normalized size = 3.03 \begin{align*} \begin{cases} \frac{\left (8 i a^{3} d e^{4 i c} e^{- 2 i d x} + 4 i a^{3} d e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{32 a^{6} d^{2}} & \text{for}\: 32 a^{6} d^{2} e^{6 i c} \neq 0 \\\frac{x \left (e^{2 i c} + 1\right ) e^{- 4 i c}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c))**3,x)

[Out]

Piecewise(((8*I*a**3*d*exp(4*I*c)*exp(-2*I*d*x) + 4*I*a**3*d*exp(2*I*c)*exp(-4*I*d*x))*exp(-6*I*c)/(32*a**6*d*
*2), Ne(32*a**6*d**2*exp(6*I*c), 0)), (x*(exp(2*I*c) + 1)*exp(-4*I*c)/(2*a**3), True))

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Giac [B]  time = 1.12068, size = 77, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{3} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-2*(tan(1/2*d*x + 1/2*c)^3 - I*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c))/(a^3*d*(tan(1/2*d*x + 1/2*c) - I
)^4)