3.165 \(\int e^t \log (1+t) \, dt\)

Optimal. Leaf size=18 \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]

[Out]

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

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Rubi [A]  time = 0.0199313, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2194, 2554, 2178} \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]

Antiderivative was successfully verified.

[In]

Int[E^t*Log[1 + t],t]

[Out]

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

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 e^t \log (1+t) \, dt &=e^t \log (1+t)-\int \frac{e^t}{1+t} \, dt\\ &=-\frac{\text{Ei}(1+t)}{e}+e^t \log (1+t)\\ \end{align*}

Mathematica [A]  time = 0.0081084, size = 18, normalized size = 1. \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[E^t*Log[1 + t],t]

[Out]

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

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Maple [A]  time = 0.02, size = 19, normalized size = 1.1 \begin{align*}{{\rm e}^{t}}\ln \left ( 1+t \right ) +{{\rm e}^{-1}}{\it Ei} \left ( 1,-1-t \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t)*ln(1+t),t)

[Out]

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

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Maxima [A]  time = 1.04043, size = 24, normalized size = 1.33 \begin{align*} e^{\left (-1\right )} E_{1}\left (-t - 1\right ) + e^{t} \log \left (t + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*log(1+t),t, algorithm="maxima")

[Out]

e^(-1)*exp_integral_e(1, -t - 1) + e^t*log(t + 1)

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Fricas [A]  time = 1.16872, size = 58, normalized size = 3.22 \begin{align*}{\left (e^{\left (t + 1\right )} \log \left (t + 1\right ) -{\rm Ei}\left (t + 1\right )\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*log(1+t),t, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{t} \log{\left (t + 1 \right )}\, dt \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*ln(1+t),t)

[Out]

Integral(exp(t)*log(t + 1), t)

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Giac [A]  time = 1.13445, size = 22, normalized size = 1.22 \begin{align*} -{\rm Ei}\left (t + 1\right ) e^{\left (-1\right )} + e^{t} \log \left (t + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t)*log(1+t),t, algorithm="giac")

[Out]

-Ei(t + 1)*e^(-1) + e^t*log(t + 1)