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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.41, size = 13, normalized size = 0.68 \[ -\frac {1}{2} \, \cos \relax (x) e^{x} + \frac {1}{2} \, e^{x} \sin \relax (x) \]

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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giac [A]  time = 0.95, size = 11, normalized size = 0.58 \[ -\frac {1}{2} \, {\left (\cos \relax (x) - \sin \relax (x)\right )} e^{x} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ -\frac {\cos \relax (x ) {\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{x} \sin \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.82, size = 11, normalized size = 0.58 \[ -\frac {1}{2} \, {\left (\cos \relax (x) - \sin \relax (x)\right )} e^{x} \]

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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mupad [B]  time = 0.00, size = 11, normalized size = 0.58 \[ -\frac {{\mathrm {e}}^x\,\left (\cos \relax (x)-\sin \relax (x)\right )}{2} \]

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 = 0.30, size = 15, normalized size = 0.79 \[ \frac {e^{x} \sin {\relax (x )}}{2} - \frac {e^{x} \cos {\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sin(x),x)

[Out]

exp(x)*sin(x)/2 - exp(x)*cos(x)/2

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