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

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

[Out]

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

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

Antiderivative was successfully verified.

[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*} \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 \begin {gather*} -\frac {e^t}{t}+\text {Ei}(t) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^t/t) + ExpIntegralEi[t]

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Maple [A]
time = 0.01, size = 16, normalized size = 1.45

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.65, size = 5, normalized size = 0.45 \begin {gather*} \Gamma \left (-1, -t\right ) \end {gather*}

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 = 0.88, size = 13, normalized size = 1.18 \begin {gather*} \frac {t {\rm Ei}\left (t\right ) - e^{t}}{t} \end {gather*}

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 = 0.42, size = 7, normalized size = 0.64 \begin {gather*} \operatorname {Ei}{\left (t \right )} - \frac {e^{t}}{t} \end {gather*}

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 = 0.44, size = 13, normalized size = 1.18 \begin {gather*} \frac {t {\rm Ei}\left (t\right ) - e^{t}}{t} \end {gather*}

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

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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