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

Optimal. Leaf size=29 \[ \frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.0147709, antiderivative size = 29, 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} \[ \frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0349311, size = 29, normalized size = 1. \[ \frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]

[Out]

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

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Maple [A]  time = 0.074, size = 23, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ad \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \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)),x)

[Out]

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

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Maxima [A]  time = 0.991109, size = 39, normalized size = 1.34 \begin{align*} \frac{2}{{\left (-i \, a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} 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)),x, algorithm="maxima")

[Out]

2/((-I*a + a*sin(d*x + c)/(cos(d*x + c) + 1))*d)

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Fricas [A]  time = 2.10801, size = 35, normalized size = 1.21 \begin{align*} \frac{i \, e^{\left (-i \, d x - i \, c\right )}}{a 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)),x, algorithm="fricas")

[Out]

I*e^(-I*d*x - I*c)/(a*d)

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Sympy [A]  time = 0.160911, size = 31, normalized size = 1.07 \begin{align*} \begin{cases} \frac{i e^{- i c} e^{- i d x}}{a d} & \text{for}\: a d e^{i c} \neq 0 \\\frac{x e^{- i c}}{a} & \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)),x)

[Out]

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

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Giac [A]  time = 1.12504, size = 28, normalized size = 0.97 \begin{align*} \frac{2}{a d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}} \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)),x, algorithm="giac")

[Out]

2/(a*d*(tan(1/2*d*x + 1/2*c) - I))