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

Optimal. Leaf size=131 \[ \frac{i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac{x}{8 a^3}+\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2} \]

[Out]

x/(8*a^3) + ((I/6)*Cos[c + d*x]^3)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3) + ((I/8)*Cos[c + d*x]^2)/(a*d*(a*
Cos[c + d*x] + I*a*Sin[c + d*x])^2) + ((I/8)*Cos[c + d*x])/(d*(a^3*Cos[c + d*x] + I*a^3*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.140537, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {3082, 8} \[ \frac{i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac{x}{8 a^3}+\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(8*a^3) + ((I/6)*Cos[c + d*x]^3)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3) + ((I/8)*Cos[c + d*x]^2)/(a*d*(a*
Cos[c + d*x] + I*a*Sin[c + d*x])^2) + ((I/8)*Cos[c + d*x])/(d*(a^3*Cos[c + d*x] + I*a^3*Sin[c + d*x]))

Rule 3082

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Simp[(b*(a*Cos[c + d*x] + b*Sin[c + d*x])^n)/(2*a*d*n*Cos[c + d*x]^n), x] + Dist[1/(2*a), Int[(a*Cos[c
+ d*x] + b*Sin[c + d*x])^(n + 1)/Cos[c + d*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && E
qQ[a^2 + b^2, 0] && LtQ[n, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx &=\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx}{2 a}\\ &=\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac{\int \frac{\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx}{4 a^2}\\ &=\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac{i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}+\frac{\int 1 \, dx}{8 a^3}\\ &=\frac{x}{8 a^3}+\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}+\frac{i \cos (c+d x)}{8 d \left (a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.117344, size = 84, normalized size = 0.64 \[ \frac{18 \sin (2 (c+d x))+9 \sin (4 (c+d x))+2 \sin (6 (c+d x))+18 i \cos (2 (c+d x))+9 i \cos (4 (c+d x))+2 i \cos (6 (c+d x))+12 c+12 d x}{96 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*c + 12*d*x + (18*I)*Cos[2*(c + d*x)] + (9*I)*Cos[4*(c + d*x)] + (2*I)*Cos[6*(c + d*x)] + 18*Sin[2*(c + d*x
)] + 9*Sin[4*(c + d*x)] + 2*Sin[6*(c + d*x)])/(96*a^3*d)

________________________________________________________________________________________

Maple [A]  time = 0.142, size = 98, normalized size = 0.8 \begin{align*}{\frac{-{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{3}}}-{\frac{{\frac{i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{1}{6\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{1}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*I/d/a^3*ln(tan(d*x+c)-I)-1/8*I/d/a^3/(tan(d*x+c)-I)^2-1/6/d/a^3/(tan(d*x+c)-I)^3+1/8/d/a^3/(tan(d*x+c)-I
)+1/16*I/d/a^3*ln(tan(d*x+c)+I)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 0.463655, size = 166, normalized size = 1.27 \begin{align*} \frac{{\left (12 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 18 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(12*d*x*e^(6*I*d*x + 6*I*c) + 18*I*e^(4*I*d*x + 4*I*c) + 9*I*e^(2*I*d*x + 2*I*c) + 2*I)*e^(-6*I*d*x - 6*I
*c)/(a^3*d)

________________________________________________________________________________________

Sympy [A]  time = 0.639258, size = 156, normalized size = 1.19 \begin{align*} \begin{cases} \frac{\left (4608 i a^{6} d^{2} e^{10 i c} e^{- 2 i d x} + 2304 i a^{6} d^{2} e^{8 i c} e^{- 4 i d x} + 512 i a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text{for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (\frac{\left (e^{6 i c} + 3 e^{4 i c} + 3 e^{2 i c} + 1\right ) e^{- 6 i c}}{8 a^{3}} - \frac{1}{8 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{x}{8 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((4608*I*a**6*d**2*exp(10*I*c)*exp(-2*I*d*x) + 2304*I*a**6*d**2*exp(8*I*c)*exp(-4*I*d*x) + 512*I*a**
6*d**2*exp(6*I*c)*exp(-6*I*d*x))*exp(-12*I*c)/(24576*a**9*d**3), Ne(24576*a**9*d**3*exp(12*I*c), 0)), (x*((exp
(6*I*c) + 3*exp(4*I*c) + 3*exp(2*I*c) + 1)*exp(-6*I*c)/(8*a**3) - 1/(8*a**3)), True)) + x/(8*a**3)

________________________________________________________________________________________

Giac [A]  time = 1.15852, size = 108, normalized size = 0.82 \begin{align*} -\frac{\frac{6 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac{6 i \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac{-11 i \, \tan \left (d x + c\right )^{3} - 45 \, \tan \left (d x + c\right )^{2} + 69 i \, \tan \left (d x + c\right ) + 51}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(6*I*log(tan(d*x + c) - I)/a^3 - 6*I*log(I*tan(d*x + c) - 1)/a^3 + (-11*I*tan(d*x + c)^3 - 45*tan(d*x +
c)^2 + 69*I*tan(d*x + c) + 51)/(a^3*(tan(d*x + c) - I)^3))/d