3.45.43 \(\int \frac {1}{20} (20-160 e^{4 x^2} x+e^{e^x} (-1-e^x x)) \, dx\)

Optimal. Leaf size=22 \[ -3-e^{4 x^2}+x-\frac {e^{e^x} x}{20} \]

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2209, 2288} \begin {gather*} -e^{4 x^2}-\frac {1}{20} e^{e^x} x+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 - 160*E^(4*x^2)*x + E^E^x*(-1 - E^x*x))/20,x]

[Out]

-E^(4*x^2) + x - (E^E^x*x)/20

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

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]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{20} \int \left (20-160 e^{4 x^2} x+e^{e^x} \left (-1-e^x x\right )\right ) \, dx\\ &=x+\frac {1}{20} \int e^{e^x} \left (-1-e^x x\right ) \, dx-8 \int e^{4 x^2} x \, dx\\ &=-e^{4 x^2}+x-\frac {e^{e^x} x}{20}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.95 \begin {gather*} -e^{4 x^2}+x-\frac {e^{e^x} x}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 - 160*E^(4*x^2)*x + E^E^x*(-1 - E^x*x))/20,x]

[Out]

-E^(4*x^2) + x - (E^E^x*x)/20

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fricas [A]  time = 0.63, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{20} \, x e^{\left (e^{x}\right )} + x - e^{\left (4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-exp(x)*x-1)*exp(exp(x))-8*x*exp(4*x^2)+1,x, algorithm="fricas")

[Out]

-1/20*x*e^(e^x) + x - e^(4*x^2)

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giac [A]  time = 0.15, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{20} \, x e^{\left (e^{x}\right )} + x - e^{\left (4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-exp(x)*x-1)*exp(exp(x))-8*x*exp(4*x^2)+1,x, algorithm="giac")

[Out]

-1/20*x*e^(e^x) + x - e^(4*x^2)

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maple [A]  time = 0.04, size = 17, normalized size = 0.77

method	result	size
default	\(x -{\mathrm e}^{4 x^{2}}-\frac {x \,{\mathrm e}^{{\mathrm e}^{x}}}{20}\)	\(17\)
norman	\(x -{\mathrm e}^{4 x^{2}}-\frac {x \,{\mathrm e}^{{\mathrm e}^{x}}}{20}\)	\(17\)
risch	\(x -{\mathrm e}^{4 x^{2}}-\frac {x \,{\mathrm e}^{{\mathrm e}^{x}}}{20}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/20*(-exp(x)*x-1)*exp(exp(x))-8*x*exp(4*x^2)+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(4*x^2)-1/20*x*exp(exp(x))

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maxima [A]  time = 0.38, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{20} \, x e^{\left (e^{x}\right )} + x - e^{\left (4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-exp(x)*x-1)*exp(exp(x))-8*x*exp(4*x^2)+1,x, algorithm="maxima")

[Out]

-1/20*x*e^(e^x) + x - e^(4*x^2)

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mupad [B]  time = 3.12, size = 16, normalized size = 0.73 \begin {gather*} x-{\mathrm {e}}^{4\,x^2}-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - (exp(exp(x))*(x*exp(x) + 1))/20 - 8*x*exp(4*x^2),x)

[Out]

x - exp(4*x^2) - (x*exp(exp(x)))/20

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sympy [A]  time = 0.21, size = 15, normalized size = 0.68 \begin {gather*} - \frac {x e^{e^{x}}}{20} + x - e^{4 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/20*(-exp(x)*x-1)*exp(exp(x))-8*x*exp(4*x**2)+1,x)

[Out]

-x*exp(exp(x))/20 + x - exp(4*x**2)

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