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

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

[Out]

-exp(t)/(1+t)+Ei(1+t)/E

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ \frac {\text {Ei}(t+1)}{e}-\frac {e^t}{t+1} \]

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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fricas [A]  time = 0.40, size = 23, normalized size = 1.21 \[ \frac {{\left ({\left (t + 1\right )} {\rm Ei}\left (t + 1\right ) - e^{\left (t + 1\right )}\right )} e^{\left (-1\right )}}{t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [B]  time = 0.01, size = 80, normalized size = 4.21 \[ \frac {{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) - {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) + e^{\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right )}}{{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} e - e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 22, normalized size = 1.16 \[ -{\mathrm e}^{-1} \Ei \left (1, -t -1\right )-\frac {{\mathrm e}^{t}}{t +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)/(t+1)^2,t)

[Out]

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

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maxima [A]  time = 0.80, size = 16, normalized size = 0.84 \[ -\frac {e^{\left (-1\right )} E_{2}\left (-t - 1\right )}{t + 1} \]

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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mupad [B]  time = 0.13, size = 17, normalized size = 0.89 \[ \mathrm {ei}\left (t+1\right )\,{\mathrm {e}}^{-1}-\frac {{\mathrm {e}}^t}{t+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)/(t + 1)^2,t)

[Out]

ei(t + 1)*exp(-1) - exp(t)/(t + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{t}}{\left (t + 1\right )^{2}}\, dt \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(t)/(t + 1)**2, t)

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