3.18 \(\int e^x \sin (x) \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{2} e^x \sin (x)-\frac{1}{2} e^x \cos (x) \]

[Out]

-(E^x*Cos[x])/2 + (E^x*Sin[x])/2

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Rubi [A]  time = 0.0074345, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4432} \[ \frac{1}{2} e^x \sin (x)-\frac{1}{2} e^x \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^x*Cos[x])/2 + (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 e^x \sin (x) \, dx &=-\frac{1}{2} e^x \cos (x)+\frac{1}{2} e^x \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.005285, size = 14, normalized size = 0.74 \[ \frac{1}{2} e^x (\sin (x)-\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^x*(-Cos[x] + Sin[x]))/2

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Maple [A]  time = 0., size = 14, normalized size = 0.7 \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(exp(x)*sin(x),x)

[Out]

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

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Maxima [A]  time = 0.928081, size = 15, normalized size = 0.79 \begin{align*} -\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(exp(x)*sin(x),x, algorithm="maxima")

[Out]

-1/2*(cos(x) - sin(x))*e^x

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Fricas [A]  time = 1.9241, size = 46, normalized size = 2.42 \begin{align*} -\frac{1}{2} \, \cos \left (x\right ) e^{x} + \frac{1}{2} \, e^{x} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x)*e^x + 1/2*e^x*sin(x)

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Sympy [A]  time = 0.282974, size = 15, normalized size = 0.79 \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(exp(x)*sin(x),x)

[Out]

exp(x)*sin(x)/2 - exp(x)*cos(x)/2

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Giac [A]  time = 1.0473, size = 15, normalized size = 0.79 \begin{align*} -\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(exp(x)*sin(x),x, algorithm="giac")

[Out]

-1/2*(cos(x) - sin(x))*e^x