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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ \text {Ei}(t)-\frac {e^t}{t} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^t/t) + ExpIntegralEi[t]

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fricas [A]  time = 0.40, size = 13, normalized size = 1.18 \[ \frac {t {\rm Ei}\relax (t) - e^{t}}{t} \]

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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giac [A]  time = 0.01, size = 13, normalized size = 1.18 \[ \frac {t {\rm Ei}\relax (t) - e^{t}}{t} \]

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

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maple [A]  time = 0.00, size = 16, normalized size = 1.45 \[ -\Ei \left (1, -t \right )-\frac {{\mathrm e}^{t}}{t} \]

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 = 0.69, size = 5, normalized size = 0.45 \[ \Gamma \left (-1, -t\right ) \]

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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mupad [B]  time = 0.02, size = 14, normalized size = 1.27 \[ -\frac {{\mathrm {e}}^t}{t}-\mathrm {expint}\left (-t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- exp(t)/t - expint(-t)

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sympy [A]  time = 0.98, size = 7, normalized size = 0.64 \[ \operatorname {Ei}{\relax (t )} - \frac {e^{t}}{t} \]

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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