3.164 \(\int \frac{e^t}{(1+t)^2} \, dt\)

Optimal. Leaf size=19 \[ \frac{\text{ExpIntegralEi}(t+1)}{e}-\frac{e^t}{t+1} \]

[Out]

-(E^t/(1 + t)) + ExpIntegralEi[1 + t]/E

_______________________________________________________________________________________

Rubi [A]  time = 0.0398312, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\text{ExpIntegralEi}(t+1)}{e}-\frac{e^t}{t+1} \]

Antiderivative was successfully verified.

[In]  Int[E^t/(1 + t)^2,t]

[Out]

-(E^t/(1 + t)) + ExpIntegralEi[1 + t]/E

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.50627, size = 12, normalized size = 0.63 \[ \frac{\operatorname{Ei}{\left (t + 1 \right )}}{e} - \frac{e^{t}}{t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(t)/(1+t)**2,t)

[Out]

exp(-1)*Ei(t + 1) - exp(t)/(t + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.011402, size = 19, normalized size = 1. \[ \frac{\text{ExpIntegralEi}(t+1)}{e}-\frac{e^t}{t+1} \]

Antiderivative was successfully verified.

[In]  Integrate[E^t/(1 + t)^2,t]

[Out]

-(E^t/(1 + t)) + ExpIntegralEi[1 + t]/E

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 22, normalized size = 1.2 \[ -{\frac{{{\rm e}^{t}}}{1+t}}-{{\rm e}^{-1}}{\it Ei} \left ( 1,-1-t \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(t)/(1+t)^2,t)

[Out]

-exp(t)/(1+t)-exp(-1)*Ei(1,-1-t)

_______________________________________________________________________________________

Maxima [A]  time = 1.43915, size = 22, normalized size = 1.16 \[ -\frac{e^{\left (-1\right )} exp_integral_e\left (2, -t - 1\right )}{t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^t/(t + 1)^2,t, algorithm="maxima")

[Out]

-e^(-1)*exp_integral_e(2, -t - 1)/(t + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.20097, size = 31, normalized size = 1.63 \[ \frac{{\left ({\left (t + 1\right )}{\rm Ei}\left (t + 1\right ) - e^{\left (t + 1\right )}\right )} e^{\left (-1\right )}}{t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^t/(t + 1)^2,t, algorithm="fricas")

[Out]

((t + 1)*Ei(t + 1) - e^(t + 1))*e^(-1)/(t + 1)

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(t)/(1+t)**2,t)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^t/(t + 1)^2,t, algorithm="giac")

[Out]

undef