3.163 \(\int \frac{e^{t^2} t}{1+t^2} \, dt\)

Optimal. Leaf size=13 \[ \frac{\text{ExpIntegralEi}\left (t^2+1\right )}{2 e} \]

[Out]

ExpIntegralEi[1 + t^2]/(2*E)

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Rubi [A]  time = 0.0654647, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6715, 2178} \[ \frac{\text{ExpIntegralEi}\left (t^2+1\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

Int[(E^t^2*t)/(1 + t^2),t]

[Out]

ExpIntegralEi[1 + t^2]/(2*E)

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), 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 \frac{e^{t^2} t}{1+t^2} \, dt &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^t}{1+t} \, dt,t,t^2\right )\\ &=\frac{\text{Ei}\left (1+t^2\right )}{2 e}\\ \end{align*}

Mathematica [A]  time = 0.0183979, size = 13, normalized size = 1. \[ \frac{\text{ExpIntegralEi}\left (t^2+1\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^t^2*t)/(1 + t^2),t]

[Out]

ExpIntegralEi[1 + t^2]/(2*E)

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Maple [A]  time = 0.006, size = 14, normalized size = 1.1 \begin{align*} -{\frac{{{\rm e}^{-1}}{\it Ei} \left ( 1,-{t}^{2}-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(t^2)*t/(t^2+1),t)

[Out]

-1/2*exp(-1)*Ei(1,-t^2-1)

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Maxima [A]  time = 1.035, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{2} \, e^{\left (-1\right )} E_{1}\left (-t^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t^2)*t/(t^2+1),t, algorithm="maxima")

[Out]

-1/2*e^(-1)*exp_integral_e(1, -t^2 - 1)

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Fricas [A]  time = 1.12247, size = 32, normalized size = 2.46 \begin{align*} \frac{1}{2} \,{\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t^2)*t/(t^2+1),t, algorithm="fricas")

[Out]

1/2*Ei(t^2 + 1)*e^(-1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{t e^{t^{2}}}{t^{2} + 1}\, dt \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t**2)*t/(t**2+1),t)

[Out]

Integral(t*exp(t**2)/(t**2 + 1), t)

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Giac [A]  time = 1.12824, size = 14, normalized size = 1.08 \begin{align*} \frac{1}{2} \,{\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(t^2)*t/(t^2+1),t, algorithm="giac")

[Out]

1/2*Ei(t^2 + 1)*e^(-1)