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

Optimal. Leaf size=31 \[ \frac{2 i}{d \sqrt{a \cos (c+d x)+i a \sin (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.017475, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3071} \[ \frac{2 i}{d \sqrt{a \cos (c+d x)+i a \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]],x]

[Out]

(2*I)/(d*Sqrt[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}{\sqrt{a \cos (c+d x)+i a \sin (c+d x)}} \, dx &=\frac{2 i}{d \sqrt{a \cos (c+d x)+i a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0350954, size = 30, normalized size = 0.97 \[ \frac{2 i}{d \sqrt{a (\cos (c+d x)+i \sin (c+d x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]],x]

[Out]

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

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Maple [A]  time = 0.029, size = 28, normalized size = 0.9 \begin{align*}{\frac{2\,i}{d}{\frac{1}{\sqrt{a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \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))^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.54175, size = 69, normalized size = 2.23 \begin{align*} \frac{2 i \, \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i}}{\sqrt{a} d \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i}} \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))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94906, size = 57, normalized size = 1.84 \begin{align*} \frac{2 i \, e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{\sqrt{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))^(1/2),x, algorithm="fricas")

[Out]

2*I*e^(-1/2*I*d*x - 1/2*I*c)/(sqrt(a)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{i a \sin{\left (c + d x \right )} + a \cos{\left (c + d x \right )}}}\, dx \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))**(1/2),x)

[Out]

Integral(1/sqrt(I*a*sin(c + d*x) + a*cos(c + d*x)), x)

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Giac [A]  time = 1.70704, size = 50, normalized size = 1.61 \begin{align*} \frac{2 i}{d \sqrt{-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i}}} \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))^(1/2),x, algorithm="giac")

[Out]

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