\(\int \frac {e^t}{t^2} \, dt\) [160]

Optimal result

Integrand size = 7, antiderivative size = 11 \[ \int \frac {e^t}{t^2} \, dt=-\frac {e^t}{t}+\operatorname {ExpIntegralEi}(t) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2208, 2209} \[ \int \frac {e^t}{t^2} \, dt=\operatorname {ExpIntegralEi}(t)-\frac {e^t}{t} \]

[In]

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

[Out]

-(E^t/t) + ExpIntegralEi[t]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t^2} \, dt=-\frac {e^t}{t}+\operatorname {ExpIntegralEi}(t) \]

[In]

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

[Out]

-(E^t/t) + ExpIntegralEi[t]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {{\mathrm e}^{t}}{t}-\operatorname {Ei}_{1}\left (-t \right )\)	\(16\)
risch	\(-\frac {{\mathrm e}^{t}}{t}-\operatorname {Ei}_{1}\left (-t \right )\)	\(16\)
meijerg	\(-\frac {1}{t}-1+\ln \left (t \right )+i \pi +\frac {2+2 t}{2 t}-\frac {{\mathrm e}^{t}}{t}-\ln \left (-t \right )-\operatorname {Ei}_{1}\left (-t \right )\)	\(44\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {e^t}{t^2} \, dt=\frac {t {\rm Ei}\left (t\right ) - e^{t}}{t} \]

[In]

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

[Out]

(t*Ei(t) - e^t)/t

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {e^t}{t^2} \, dt=\operatorname {Ei}{\left (t \right )} - \frac {e^{t}}{t} \]

[In]

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

[Out]

Ei(t) - exp(t)/t

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.45 \[ \int \frac {e^t}{t^2} \, dt=\Gamma \left (-1, -t\right ) \]

[In]

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

[Out]

gamma(-1, -t)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {e^t}{t^2} \, dt=\frac {t {\rm Ei}\left (t\right ) - e^{t}}{t} \]

[In]

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

[Out]

(t*Ei(t) - e^t)/t

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \frac {e^t}{t^2} \, dt=-\frac {{\mathrm {e}}^t}{t}-\mathrm {expint}\left (-t\right ) \]

[In]

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

[Out]

- exp(t)/t - expint(-t)