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

Optimal. Leaf size=41 \[ -\frac{1}{2} e^{-x} \sin (x)+\frac{1}{2} e^x \sin (x)-\frac{1}{2} e^{-x} \cos (x)-\frac{1}{2} e^x \cos (x) \]

[Out]

-Cos[x]/(2*E^x) - (E^x*Cos[x])/2 - Sin[x]/(2*E^x) + (E^x*Sin[x])/2

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Rubi [A]  time = 0.0252271, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {4432} \[ -\frac{1}{2} e^{-x} \sin (x)+\frac{1}{2} e^x \sin (x)-\frac{1}{2} e^{-x} \cos (x)-\frac{1}{2} e^x \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/E^x + E^x*Sin[x],x]

[Out]

-Cos[x]/(2*E^x) - (E^x*Cos[x])/2 - Sin[x]/(2*E^x) + (E^x*Sin[x])/2

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

Mathematica [A]  time = 0.0646951, size = 33, normalized size = 0.8 \[ -\frac{1}{2} e^x \left (e^{-2 x}-1\right ) \sin (x)-\frac{1}{2} e^x \left (e^{-2 x}+1\right ) \cos (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/E^x + E^x*Sin[x],x]

[Out]

-(E^x*(1 + E^(-2*x))*Cos[x])/2 - (E^x*(-1 + E^(-2*x))*Sin[x])/2

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Maple [A]  time = 0.007, size = 30, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{x}}\cos \left ( x \right ) }{2}}+{\frac{{{\rm e}^{x}}\sin \left ( x \right ) }{2}}-{\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)+exp(x)*sin(x),x)

[Out]

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

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Maxima [A]  time = 1.03471, size = 31, normalized size = 0.76 \begin{align*} -\frac{1}{2} \,{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac{1}{2} \,{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.463393, size = 84, normalized size = 2.05 \begin{align*} -\frac{1}{2} \,{\left (\cos \left (x\right ) e^{\left (2 \, x\right )} -{\left (e^{\left (2 \, x\right )} - 1\right )} \sin \left (x\right ) + \cos \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)+exp(x)*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.60042, size = 32, normalized size = 0.78 \begin{align*} \frac{e^{x} \sin{\left (x \right )}}{2} - \frac{e^{x} \cos{\left (x \right )}}{2} - \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)+exp(x)*sin(x),x)

[Out]

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

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Giac [A]  time = 1.1592, size = 31, normalized size = 0.76 \begin{align*} -\frac{1}{2} \,{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac{1}{2} \,{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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