3.252 \(\int (a \cos (c+d x)+i a \sin (c+d x)) \, dx\)

Optimal. Leaf size=26 \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0140323, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2637, 2638} \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0118994, size = 51, normalized size = 1.96 \[ \frac{i a \sin (c) \sin (d x)}{d}-\frac{i a \cos (c) \cos (d x)}{d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 26, normalized size = 1. \begin{align*}{\frac{-ia\cos \left ( dx+c \right ) }{d}}+{\frac{a\sin \left ( dx+c \right ) }{d}} \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),x)

[Out]

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

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Maxima [A]  time = 0.98745, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )}{d} + \frac{a \sin \left (d x + 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.16157, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a e^{\left (i \, d x + 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),x, algorithm="fricas")

[Out]

-I*a*e^(I*d*x + I*c)/d

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Sympy [A]  time = 0.144124, size = 26, normalized size = 1. \begin{align*} \begin{cases} - \frac{i a e^{i c} e^{i d x}}{d} & \text{for}\: d \neq 0 \\a x e^{i c} & \text{otherwise} \end{cases} \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),x)

[Out]

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

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Giac [A]  time = 1.12907, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )}{d} + \frac{a \sin \left (d x + 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),x, algorithm="giac")

[Out]

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