3.32 \(\int e^{2 t} \sin (3 t) \, dt\)

Optimal. Leaf size=27 \[ \frac{2}{13} e^{2 t} \sin (3 t)-\frac{3}{13} e^{2 t} \cos (3 t) \]

[Out]

(-3*E^(2*t)*Cos[3*t])/13 + (2*E^(2*t)*Sin[3*t])/13

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Rubi [A]  time = 0.0206872, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2}{13} e^{2 t} \sin (3 t)-\frac{3}{13} e^{2 t} \cos (3 t) \]

Antiderivative was successfully verified.

[In]  Int[E^(2*t)*Sin[3*t],t]

[Out]

(-3*E^(2*t)*Cos[3*t])/13 + (2*E^(2*t)*Sin[3*t])/13

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Rubi in Sympy [A]  time = 1.44696, size = 26, normalized size = 0.96 \[ \frac{2 e^{2 t} \sin{\left (3 t \right )}}{13} - \frac{3 e^{2 t} \cos{\left (3 t \right )}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*t)*sin(3*t),t)

[Out]

2*exp(2*t)*sin(3*t)/13 - 3*exp(2*t)*cos(3*t)/13

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Mathematica [A]  time = 0.0288039, size = 22, normalized size = 0.81 \[ \frac{1}{13} e^{2 t} (2 \sin (3 t)-3 \cos (3 t)) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*t)*Sin[3*t],t]

[Out]

(E^(2*t)*(-3*Cos[3*t] + 2*Sin[3*t]))/13

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Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \[ -{\frac{3\,{{\rm e}^{2\,t}}\cos \left ( 3\,t \right ) }{13}}+{\frac{2\,{{\rm e}^{2\,t}}\sin \left ( 3\,t \right ) }{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*t)*sin(3*t),t)

[Out]

-3/13*exp(2*t)*cos(3*t)+2/13*exp(2*t)*sin(3*t)

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Maxima [A]  time = 1.40284, size = 26, normalized size = 0.96 \[ -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, t\right ) - 2 \, \sin \left (3 \, t\right )\right )} e^{\left (2 \, t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*t)*sin(3*t),t, algorithm="maxima")

[Out]

-1/13*(3*cos(3*t) - 2*sin(3*t))*e^(2*t)

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Fricas [A]  time = 0.217567, size = 28, normalized size = 1.04 \[ -\frac{3}{13} \, \cos \left (3 \, t\right ) e^{\left (2 \, t\right )} + \frac{2}{13} \, e^{\left (2 \, t\right )} \sin \left (3 \, t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*t)*sin(3*t),t, algorithm="fricas")

[Out]

-3/13*cos(3*t)*e^(2*t) + 2/13*e^(2*t)*sin(3*t)

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Sympy [A]  time = 0.373493, size = 26, normalized size = 0.96 \[ \frac{2 e^{2 t} \sin{\left (3 t \right )}}{13} - \frac{3 e^{2 t} \cos{\left (3 t \right )}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*t)*sin(3*t),t)

[Out]

2*exp(2*t)*sin(3*t)/13 - 3*exp(2*t)*cos(3*t)/13

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GIAC/XCAS [A]  time = 0.201499, size = 26, normalized size = 0.96 \[ -\frac{1}{13} \,{\left (3 \, \cos \left (3 \, t\right ) - 2 \, \sin \left (3 \, t\right )\right )} e^{\left (2 \, t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*t)*sin(3*t),t, algorithm="giac")

[Out]

-1/13*(3*cos(3*t) - 2*sin(3*t))*e^(2*t)