\(\int \frac {e^{t^2} t}{1+t^2} \, dt\) [163]

Optimal result

Integrand size = 14, antiderivative size = 13 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {\operatorname {ExpIntegralEi}\left (1+t^2\right )}{2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6847, 2209} \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {\operatorname {ExpIntegralEi}\left (t^2+1\right )}{2 e} \]

[In]

Int[(E^t^2*t)/(1 + t^2),t]

[Out]

ExpIntegralEi[1 + t^2]/(2*E)

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]

Rule 6847

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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {e^t}{1+t} \, dt,t,t^2\right ) \\ & = \frac {\operatorname {ExpIntegralEi}\left (1+t^2\right )}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {\operatorname {ExpIntegralEi}\left (1+t^2\right )}{2 e} \]

[In]

Integrate[(E^t^2*t)/(1 + t^2),t]

[Out]

ExpIntegralEi[1 + t^2]/(2*E)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\frac {{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-t^{2}-1\right )}{2}\)	\(14\)
default	\(-\frac {{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-t^{2}-1\right )}{2}\)	\(14\)
risch	\(-\frac {{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-t^{2}-1\right )}{2}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {1}{2} \, {\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \]

[In]

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

[Out]

1/2*Ei(t^2 + 1)*e^(-1)

Sympy [F]

\[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\int \frac {t e^{t^{2}}}{t^{2} + 1}\, dt \]

[In]

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

[Out]

Integral(t*exp(t**2)/(t**2 + 1), t)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=-\frac {1}{2} \, e^{\left (-1\right )} E_{1}\left (-t^{2} - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {1}{2} \, {\rm Ei}\left (t^{2} + 1\right ) e^{\left (-1\right )} \]

[In]

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

[Out]

1/2*Ei(t^2 + 1)*e^(-1)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {e^{t^2} t}{1+t^2} \, dt=\frac {{\mathrm {e}}^{-1}\,\mathrm {ei}\left (t^2+1\right )}{2} \]

[In]

int((t*exp(t^2))/(t^2 + 1),t)

[Out]

(exp(-1)*ei(t^2 + 1))/2