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

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

[Out]

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

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Rubi [A]  time = 0.0160164, 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} \[ -\frac{2 i \sqrt{a \cos (c+d x)+i a \sin (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0220753, size = 30, normalized size = 0.97 \[ -\frac{2 i \sqrt{a (\cos (c+d x)+i \sin (c+d x))}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 28, normalized size = 0.9 \begin{align*}{\frac{-2\,i}{d}\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((a*cos(d*x+c)+I*a*sin(d*x+c))^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.54232, size = 69, normalized size = 2.23 \begin{align*} -\frac{2 i \, \sqrt{a} \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i}}{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((a*cos(d*x+c)+I*a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.01866, size = 54, normalized size = 1.74 \begin{align*} -\frac{2 i \, \sqrt{a} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.6811, size = 23, normalized size = 0.74 \begin{align*} -\frac{2 i \, \sqrt{a} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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