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

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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]

-1/2*(Cos[x] + Sin[x])/E^x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-x} \sin (x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.88, size = 17, normalized size = 0.74 \[ -\frac {1}{2} \, \cos \relax (x) e^{\left (-x\right )} - \frac {1}{2} \, e^{\left (-x\right )} \sin \relax (x) \]

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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giac [A]  time = 0.94, size = 11, normalized size = 0.48 \[ -\frac {1}{2} \, {\left (\cos \relax (x) + \sin \relax (x)\right )} e^{\left (-x\right )} \]

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)

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maple [A]  time = 0.04, size = 18, normalized size = 0.78

method	result	size
default	\(-\frac {{\mathrm e}^{-x} \cos \relax (x )}{2}-\frac {{\mathrm e}^{-x} \sin \relax (x )}{2}\)	\(18\)
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	\(-\frac {{\mathrm e}^{\left (-1+i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 11, normalized size = 0.48 \[ -\frac {1}{2} \, {\left (\cos \relax (x) + \sin \relax (x)\right )} e^{\left (-x\right )} \]

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.48 \[ -\frac {{\mathrm {e}}^{-x}\,\left (\cos \relax (x)+\sin \relax (x)\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(-x)*(cos(x) + sin(x)))/2

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sympy [A]  time = 0.46, size = 17, normalized size = 0.74 \[ - \frac {e^{- x} \sin {\relax (x )}}{2} - \frac {e^{- x} \cos {\relax (x )}}{2} \]

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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