3.255 \(\int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=31 \[ \frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3071} \[ \frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3071

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

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 31, normalized size = 1.00 \[ \frac {i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.62, size = 17, normalized size = 0.55 \[ \frac {i \, e^{\left (-3 i \, d x - 3 i \, c\right )}}{3 \, a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/3*I*e^(-3*I*d*x - 3*I*c)/(a^3*d)

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giac [A]  time = 0.15, size = 36, normalized size = 1.16 \[ \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{3 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.43, size = 57, normalized size = 1.84 \[ \frac {\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}-\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {4 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x)

[Out]

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

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maxima [A]  time = 0.63, size = 29, normalized size = 0.94 \[ \frac {i \, \cos \left (3 \, d x + 3 \, c\right ) + \sin \left (3 \, d x + 3 \, c\right )}{3 \, a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.46, size = 68, normalized size = 2.19 \[ -\frac {2\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,a^3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(c + d*x) + a*sin(c + d*x)*1i)^3,x)

[Out]

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

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sympy [A]  time = 0.14, size = 46, normalized size = 1.48 \[ \begin {cases} \frac {i e^{- 3 i c} e^{- 3 i d x}}{3 a^{3} d} & \text {for}\: 3 a^{3} d e^{3 i c} \neq 0 \\\frac {x e^{- 3 i c}}{a^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))**3,x)

[Out]

Piecewise((I*exp(-3*I*c)*exp(-3*I*d*x)/(3*a**3*d), Ne(3*a**3*d*exp(3*I*c), 0)), (x*exp(-3*I*c)/a**3, True))

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