\(\int e^{2 x^2} x \cos (2 x^2) \, dx\) [61]

Optimal result

Integrand size = 15, antiderivative size = 35 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} e^{2 x^2} \cos \left (2 x^2\right )+\frac {1}{8} e^{2 x^2} \sin \left (2 x^2\right ) \]

[Out]

1/8*exp(2*x^2)*cos(2*x^2)+1/8*exp(2*x^2)*sin(2*x^2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6847, 4518} \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} e^{2 x^2} \sin \left (2 x^2\right )+\frac {1}{8} e^{2 x^2} \cos \left (2 x^2\right ) \]

[In]

Int[E^(2*x^2)*x*Cos[2*x^2],x]

[Out]

(E^(2*x^2)*Cos[2*x^2])/8 + (E^(2*x^2)*Sin[2*x^2])/8

Rule 4518

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

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 e^{2 x} \cos (2 x) \, dx,x,x^2\right ) \\ & = \frac {1}{8} e^{2 x^2} \cos \left (2 x^2\right )+\frac {1}{8} e^{2 x^2} \sin \left (2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} e^{2 x^2} \left (\cos \left (2 x^2\right )+\sin \left (2 x^2\right )\right ) \]

[In]

Integrate[E^(2*x^2)*x*Cos[2*x^2],x]

[Out]

(E^(2*x^2)*(Cos[2*x^2] + Sin[2*x^2]))/8

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{2 x^{2}} \left (\cos \left (2 x^{2}\right )+\sin \left (2 x^{2}\right )\right )}{8}\)	\(22\)
derivativedivides	\(\frac {{\mathrm e}^{2 x^{2}} \cos \left (2 x^{2}\right )}{8}+\frac {{\mathrm e}^{2 x^{2}} \sin \left (2 x^{2}\right )}{8}\)	\(30\)
default	\(\frac {{\mathrm e}^{2 x^{2}} \cos \left (2 x^{2}\right )}{8}+\frac {{\mathrm e}^{2 x^{2}} \sin \left (2 x^{2}\right )}{8}\)	\(30\)
risch	\(\frac {{\mathrm e}^{\left (2+2 i\right ) x^{2}}}{16}-\frac {i {\mathrm e}^{\left (2+2 i\right ) x^{2}}}{16}+\frac {{\mathrm e}^{\left (2-2 i\right ) x^{2}}}{16}+\frac {i {\mathrm e}^{\left (2-2 i\right ) x^{2}}}{16}\)	\(44\)
norman	\(\frac {\frac {{\mathrm e}^{2 x^{2}} \tan \left (x^{2}\right )}{4}-\frac {{\mathrm e}^{2 x^{2}} \tan \left (x^{2}\right )^{2}}{8}+\frac {{\mathrm e}^{2 x^{2}}}{8}}{1+\tan \left (x^{2}\right )^{2}}\)	\(47\)

[In]

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

[Out]

1/8*exp(2*x^2)*(cos(2*x^2)+sin(2*x^2))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} \, \cos \left (2 \, x^{2}\right ) e^{\left (2 \, x^{2}\right )} + \frac {1}{8} \, e^{\left (2 \, x^{2}\right )} \sin \left (2 \, x^{2}\right ) \]

[In]

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

[Out]

1/8*cos(2*x^2)*e^(2*x^2) + 1/8*e^(2*x^2)*sin(2*x^2)

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {e^{2 x^{2}} \sin {\left (2 x^{2} \right )}}{8} + \frac {e^{2 x^{2}} \cos {\left (2 x^{2} \right )}}{8} \]

[In]

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

[Out]

exp(2*x**2)*sin(2*x**2)/8 + exp(2*x**2)*cos(2*x**2)/8

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} \, \cos \left (2 \, x^{2}\right ) e^{\left (2 \, x^{2}\right )} + \frac {1}{8} \, e^{\left (2 \, x^{2}\right )} \sin \left (2 \, x^{2}\right ) \]

[In]

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

[Out]

1/8*cos(2*x^2)*e^(2*x^2) + 1/8*e^(2*x^2)*sin(2*x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {1}{8} \, {\left (\cos \left (2 \, x^{2}\right ) + \sin \left (2 \, x^{2}\right )\right )} e^{\left (2 \, x^{2}\right )} \]

[In]

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

[Out]

1/8*(cos(2*x^2) + sin(2*x^2))*e^(2*x^2)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int e^{2 x^2} x \cos \left (2 x^2\right ) \, dx=\frac {{\mathrm {e}}^{2\,x^2}\,\left (\cos \left (2\,x^2\right )+\sin \left (2\,x^2\right )\right )}{8} \]

[In]

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

[Out]

(exp(2*x^2)*(cos(2*x^2) + sin(2*x^2)))/8