\(\int e^{-x} \sin (x) \, dx\) [77]

Optimal result

Integrand size = 8, antiderivative size = 23 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^{-x} \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4517} \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} \sin (x)-\frac {1}{2} e^{-x} \cos (x) \]

[In]

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

[Out]

-1/2*Cos[x]/E^x - Sin[x]/(2*E^x)

Rule 4517

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*} \text {integral}& = -\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^{-x} \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} (\cos (x)+\sin (x)) \]

[In]

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

[Out]

-1/2*(Cos[x] + Sin[x])/E^x

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52

method	result	size
parallelrisch	\(-\frac {\left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{-x}}{2}\)	\(12\)
default	\(-\frac {{\mathrm e}^{-x} \cos \left (x \right )}{2}-\frac {{\mathrm e}^{-x} \sin \left (x \right )}{2}\)	\(18\)
norman	\(\frac {\left (-\frac {1}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-x}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(32\)
risch	\(-\frac {{\mathrm e}^{\left (-1+i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}\)	\(36\)

[In]

int(sin(x)/exp(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right ) e^{\left (-x\right )} - \frac {1}{2} \, e^{\left (-x\right )} \sin \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-x} \sin (x) \, dx=- \frac {e^{- x} \sin {\left (x \right )}}{2} - \frac {e^{- x} \cos {\left (x \right )}}{2} \]

[In]

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

[Out]

-exp(-x)*sin(x)/2 - exp(-x)*cos(x)/2

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \]

[In]

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

[Out]

-1/2*(cos(x) + sin(x))*e^(-x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \]

[In]

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

[Out]

-1/2*(cos(x) + sin(x))*e^(-x)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {{\mathrm {e}}^{-x}\,\left (\cos \left (x\right )+\sin \left (x\right )\right )}{2} \]

[In]

int(exp(-x)*sin(x),x)

[Out]

-(exp(-x)*(cos(x) + sin(x)))/2