3.77 \(\int e^{-x} \sin (x) \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{2} e^{-x} \sin (x)-\frac{1}{2} e^{-x} \cos (x) \]

[Out]

-Cos[x]/(2*E^x) - Sin[x]/(2*E^x)

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Rubi [A]  time = 0.0084535, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4432} \[ -\frac{1}{2} e^{-x} \sin (x)-\frac{1}{2} e^{-x} \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/E^x,x]

[Out]

-Cos[x]/(2*E^x) - Sin[x]/(2*E^x)

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int e^{-x} \sin (x) \, dx &=-\frac{1}{2} e^{-x} \cos (x)-\frac{1}{2} e^{-x} \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0123571, size = 14, normalized size = 0.61 \[ -\frac{1}{2} e^{-x} (\sin (x)+\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/E^x,x]

[Out]

-(Cos[x] + Sin[x])/(2*E^x)

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Maple [A]  time = 0.004, size = 18, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{-x}}\cos \left ( x \right ) }{2}}-{\frac{{{\rm e}^{-x}}\sin \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/exp(x),x)

[Out]

-1/2*exp(-x)*cos(x)-1/2*exp(-x)*sin(x)

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Maxima [A]  time = 0.938448, size = 15, normalized size = 0.65 \begin{align*} -\frac{1}{2} \,{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/exp(x),x, algorithm="maxima")

[Out]

-1/2*(cos(x) + sin(x))*e^(-x)

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Fricas [A]  time = 2.15851, size = 54, normalized size = 2.35 \begin{align*} -\frac{1}{2} \, \cos \left (x\right ) e^{\left (-x\right )} - \frac{1}{2} \, e^{\left (-x\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/exp(x),x, algorithm="fricas")

[Out]

-1/2*cos(x)*e^(-x) - 1/2*e^(-x)*sin(x)

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Sympy [A]  time = 0.4605, size = 17, normalized size = 0.74 \begin{align*} - \frac{e^{- x} \sin{\left (x \right )}}{2} - \frac{e^{- x} \cos{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/exp(x),x)

[Out]

-exp(-x)*sin(x)/2 - exp(-x)*cos(x)/2

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Giac [A]  time = 1.05146, size = 15, normalized size = 0.65 \begin{align*} -\frac{1}{2} \,{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/exp(x),x, algorithm="giac")

[Out]

-1/2*(cos(x) + sin(x))*e^(-x)