3.160 \(\int \frac{e^t}{t^2} \, dt\)

Optimal. Leaf size=11 \[ \text{ExpIntegralEi}(t)-\frac{e^t}{t} \]

[Out]

-(E^t/t) + ExpIntegralEi[t]

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Rubi [A]  time = 0.0196118, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2177, 2178} \[ \text{ExpIntegralEi}(t)-\frac{e^t}{t} \]

Antiderivative was successfully verified.

[In]

Int[E^t/t^2,t]

[Out]

-(E^t/t) + ExpIntegralEi[t]

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

Mathematica [A]  time = 0.0073473, size = 11, normalized size = 1. \[ \text{ExpIntegralEi}(t)-\frac{e^t}{t} \]

Antiderivative was successfully verified.

[In]

Integrate[E^t/t^2,t]

[Out]

-(E^t/t) + ExpIntegralEi[t]

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Maple [A]  time = 0.002, size = 16, normalized size = 1.5 \begin{align*} -{\frac{{{\rm e}^{t}}}{t}}-{\it Ei} \left ( 1,-t \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02161, size = 7, normalized size = 0.64 \begin{align*} \Gamma \left (-1, -t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1, -t)

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Fricas [A]  time = 1.04303, size = 26, normalized size = 2.36 \begin{align*} \frac{t{\rm Ei}\left (t\right ) - e^{t}}{t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t*Ei(t) - e^t)/t

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Sympy [A]  time = 0.965294, size = 7, normalized size = 0.64 \begin{align*} \operatorname{Ei}{\left (t \right )} - \frac{e^{t}}{t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(t) - exp(t)/t

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Giac [A]  time = 1.10232, size = 18, normalized size = 1.64 \begin{align*} \frac{t{\rm Ei}\left (t\right ) - e^{t}}{t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t*Ei(t) - e^t)/t