3.2.64 \(\int \frac {e^t}{(1+t)^2} \, dt\) [164]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {2208, 2209} \begin {gather*} \frac {\text {Ei}(t+1)}{e}-\frac {e^t}{t+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^t/(1 + t)^2,t]

[Out]

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

Rule 2208

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] &&  !TrueQ[$UseGamm
a]

Rule 2209

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

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 = 0.03, size = 19, normalized size = 1.00 \begin {gather*} -\frac {e^t}{1+t}+\frac {\text {Ei}(1+t)}{e} \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[E^t/(t + 1)^2,t]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.06, size = 22, normalized size = 1.16

method	result	size
default	\(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \expIntegral \left (1, -1-t \right )\)	\(22\)
risch	\(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \expIntegral \left (1, -1-t \right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)/(1+t)^2,t,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.37, size = 16, normalized size = 0.84 \begin {gather*} -\frac {e^{\left (-1\right )} E_{2}\left (-t - 1\right )}{t + 1} \end {gather*}

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 = 0.32, size = 23, normalized size = 1.21 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{t}}{\left (t + 1\right )^{2}}\, dt \end {gather*}

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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Giac [A]
time = 0.00, size = 26, normalized size = 1.37 \begin {gather*} \frac {t \mathrm {Ei}\left (t+1\right )+\mathrm {Ei}\left (t+1\right )-\mathrm {e} \mathrm {e}^{t}}{t \mathrm {e}+\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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