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

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

[Out]

1/2*Ei(t^2+1)/E

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Rubi [A]  time = 0.07, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 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

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]

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*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 1.00 \[ \frac {\text {Ei}\left (t^2+1\right )}{2 e} \]

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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fricas [A]  time = 0.39, size = 10, normalized size = 0.77 \[ \frac {1}{2} \, {\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \]

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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giac [A]  time = 0.01, size = 10, normalized size = 0.77 \[ \frac {1}{2} \, {\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \]

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)

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maple [A]  time = 0.01, size = 14, normalized size = 1.08 \[ -\frac {{\mathrm e}^{-1} \Ei \left (1, -t^{2}-1\right )}{2} \]

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 = 0.79, size = 13, normalized size = 1.00 \[ -\frac {1}{2} \, e^{\left (-1\right )} E_{1}\left (-t^{2} - 1\right ) \]

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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mupad [B]  time = 0.12, size = 10, normalized size = 0.77 \[ \frac {{\mathrm {e}}^{-1}\,\mathrm {ei}\left (t^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((t*exp(t^2))/(t^2 + 1),t)

[Out]

(exp(-1)*ei(t^2 + 1))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {t e^{t^{2}}}{t^{2} + 1}\, dt \]

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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