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

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Rubi [A]  time = 0.0233803, 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, Rules used = {2177, 2178} \[ \frac{\text{ExpIntegralEi}(t+1)}{e}-\frac{e^t}{t+1} \]

Antiderivative was successfully verified.

[In]

Int[E^t/(1 + t)^2,t]

[Out]

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

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^t}{(1+t)^2} \, dt &=-\frac{e^t}{1+t}+\int \frac{e^t}{1+t} \, dt\\ &=-\frac{e^t}{1+t}+\frac{\text{Ei}(1+t)}{e}\\ \end{align*}

Mathematica [A]  time = 0.019306, 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

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Maple [A]  time = 0.003, size = 22, normalized size = 1.2 \begin{align*} -{\frac{{{\rm e}^{t}}}{1+t}}-{{\rm e}^{-1}}{\it Ei} \left ( 1,-1-t \right ) \end{align*}

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)

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Maxima [A]  time = 1.03167, size = 22, normalized size = 1.16 \begin{align*} -\frac{e^{\left (-1\right )} E_{2}\left (-t - 1\right )}{t + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)/(1+t)^2,t, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.09991, size = 65, normalized size = 3.42 \begin{align*} \frac{{\left ({\left (t + 1\right )}{\rm Ei}\left (t + 1\right ) - e^{\left (t + 1\right )}\right )} e^{\left (-1\right )}}{t + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)/(1+t)^2,t, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)/(1+t)**2,t)

[Out]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{t}}{{\left (t + 1\right )}^{2}}\,{d t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)/(1+t)^2,t, algorithm="giac")

[Out]

integrate(e^t/(t + 1)^2, t)