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]

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

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Rubi [A]  time = 0.0148451, antiderivative size = 31, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0486514, size = 31, normalized size = 1. \[ \frac{i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4} \]

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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Maple [A]  time = 0.115, size = 36, normalized size = 1.2 \begin{align*}{\frac{1}{d{a}^{4}} \left ({\frac{-i}{ \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}- \left ( \tan \left ( dx+c \right ) -i \right ) ^{-1} \right ) } \end{align*}

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 = 1.03061, size = 39, normalized size = 1.26 \begin{align*} \frac{i \, \cos \left (4 \, d x + 4 \, c\right ) + \sin \left (4 \, d x + 4 \, c\right )}{4 \, a^{4} d} \end{align*}

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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Fricas [A]  time = 2.06525, size = 49, normalized size = 1.58 \begin{align*} \frac{i \, e^{\left (-4 i \, d x - 4 i \, c\right )}}{4 \, a^{4} d} \end{align*}

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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Sympy [A]  time = 0.194294, size = 46, normalized size = 1.48 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.14093, size = 59, normalized size = 1.9 \begin{align*} -\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}} \end{align*}

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)