\(\int e^{x^2} x \, dx\) [167]

Optimal result

Integrand size = 7, antiderivative size = 9 \[ \int e^{x^2} x \, dx=\frac {e^{x^2}}{2} \]

[Out]

1/2*exp(x^2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2240} \[ \int e^{x^2} x \, dx=\frac {e^{x^2}}{2} \]

[In]

Int[E^x^2*x,x]

[Out]

E^x^2/2

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{x^2}}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int e^{x^2} x \, dx=\frac {e^{x^2}}{2} \]

[In]

Integrate[E^x^2*x,x]

[Out]

E^x^2/2

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78

method	result	size
gosper	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
derivativedivides	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
default	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
norman	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
risch	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
parallelrisch	\(\frac {{\mathrm e}^{x^{2}}}{2}\)	\(7\)
meijerg	\(-\frac {1}{2}+\frac {{\mathrm e}^{x^{2}}}{2}\)	\(9\)
parts	\(\frac {\operatorname {erfi}\left (x \right ) \sqrt {\pi }\, x}{2}-\frac {\sqrt {\pi }\, \left (x \,\operatorname {erfi}\left (x \right )-\frac {{\mathrm e}^{x^{2}}}{\sqrt {\pi }}\right )}{2}\)	\(29\)

[In]

int(exp(x^2)*x,x,method=_RETURNVERBOSE)

[Out]

1/2*exp(x^2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int e^{x^2} x \, dx=\frac {1}{2} \, e^{\left (x^{2}\right )} \]

[In]

integrate(exp(x^2)*x,x, algorithm="fricas")

[Out]

1/2*e^(x^2)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int e^{x^2} x \, dx=\frac {e^{x^{2}}}{2} \]

[In]

integrate(exp(x**2)*x,x)

[Out]

exp(x**2)/2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int e^{x^2} x \, dx=\frac {1}{2} \, e^{\left (x^{2}\right )} \]

[In]

integrate(exp(x^2)*x,x, algorithm="maxima")

[Out]

1/2*e^(x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int e^{x^2} x \, dx=\frac {1}{2} \, e^{\left (x^{2}\right )} \]

[In]

integrate(exp(x^2)*x,x, algorithm="giac")

[Out]

1/2*e^(x^2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int e^{x^2} x \, dx=\frac {{\mathrm {e}}^{x^2}}{2} \]

[In]

int(x*exp(x^2),x)

[Out]

exp(x^2)/2