3.256 \(\int \frac {1}{(a \cos (c+d x)+i a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=31 \[ \frac {i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4} \]

[Out]

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

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Rubi [A]  time = 0.01, 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} \[ \frac {i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-4),x]

[Out]

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

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))^4} \, dx &=\frac {i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 31, normalized size = 1.00 \[ \frac {i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-4),x]

[Out]

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

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fricas [A]  time = 1.33, size = 17, normalized size = 0.55 \[ \frac {i \, e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, a^{4} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/4*I*e^(-4*I*d*x - 4*I*c)/(a^4*d)

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giac [A]  time = 0.16, size = 44, normalized size = 1.42 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{4} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.42, size = 36, normalized size = 1.16 \[ \frac {-\frac {i}{\left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{\tan \left (d x +c \right )-i}}{d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^4,x)

[Out]

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

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maxima [A]  time = 0.57, size = 29, normalized size = 0.94 \[ \frac {i \, \cos \left (4 \, d x + 4 \, c\right ) + \sin \left (4 \, d x + 4 \, c\right )}{4 \, a^{4} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.56, size = 91, normalized size = 2.94 \[ -\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,1{}\mathrm {i}-\mathrm {i}\right )}{a^4\,d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,1{}\mathrm {i}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,6{}\mathrm {i}-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(c + d*x) + a*sin(c + d*x)*1i)^4,x)

[Out]

-(2*tan(c/2 + (d*x)/2)*(tan(c/2 + (d*x)/2)^2*1i - 1i))/(a^4*d*(4*tan(c/2 + (d*x)/2)^3 - tan(c/2 + (d*x)/2)^2*6
i - 4*tan(c/2 + (d*x)/2) + tan(c/2 + (d*x)/2)^4*1i + 1i))

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sympy [A]  time = 0.15, size = 46, normalized size = 1.48 \[ \begin {cases} \frac {i e^{- 4 i c} e^{- 4 i d x}}{4 a^{4} d} & \text {for}\: 4 a^{4} d e^{4 i c} \neq 0 \\\frac {x e^{- 4 i c}}{a^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(d*x+c)+I*a*sin(d*x+c))**4,x)

[Out]

Piecewise((I*exp(-4*I*c)*exp(-4*I*d*x)/(4*a**4*d), Ne(4*a**4*d*exp(4*I*c), 0)), (x*exp(-4*I*c)/a**4, True))

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