3.31 \(\int e^t t^3 \, dt\)

Optimal. Leaf size=27 \[ e^t t^3-3 e^t t^2+6 e^t t-6 e^t \]

[Out]

-6*E^t + 6*E^t*t - 3*E^t*t^2 + E^t*t^3

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Rubi [A]  time = 0.047074, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ e^t t^3-3 e^t t^2+6 e^t t-6 e^t \]

Antiderivative was successfully verified.

[In]  Int[E^t*t^3,t]

[Out]

-6*E^t + 6*E^t*t - 3*E^t*t^2 + E^t*t^3

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Rubi in Sympy [A]  time = 2.64576, size = 26, normalized size = 0.96 \[ t^{3} e^{t} - 3 t^{2} e^{t} + 6 t e^{t} - 6 e^{t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(t)*t**3,t)

[Out]

t**3*exp(t) - 3*t**2*exp(t) + 6*t*exp(t) - 6*exp(t)

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Mathematica [A]  time = 0.00284049, size = 17, normalized size = 0.63 \[ e^t \left (t^3-3 t^2+6 t-6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^t*t^3,t]

[Out]

E^t*(-6 + 6*t - 3*t^2 + t^3)

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Maple [A]  time = 0.004, size = 17, normalized size = 0.6 \[ \left ({t}^{3}-3\,{t}^{2}+6\,t-6 \right ){{\rm e}^{t}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(t)*t^3,t)

[Out]

(t^3-3*t^2+6*t-6)*exp(t)

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Maxima [A]  time = 1.40972, size = 22, normalized size = 0.81 \[{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t

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Fricas [A]  time = 0.205059, size = 22, normalized size = 0.81 \[{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t

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Sympy [A]  time = 0.06951, size = 15, normalized size = 0.56 \[ \left (t^{3} - 3 t^{2} + 6 t - 6\right ) e^{t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(t)*t**3,t)

[Out]

(t**3 - 3*t**2 + 6*t - 6)*exp(t)

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GIAC/XCAS [A]  time = 0.198196, size = 22, normalized size = 0.81 \[{\left (t^{3} - 3 \, t^{2} + 6 \, t - 6\right )} e^{t} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(t^3 - 3*t^2 + 6*t - 6)*e^t